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The algebra of conditional sets and the concepts ofconditional topology and compactnessSamuel Drapeaua,1,∗, Asgar Jamneshanb,2,†,§, Martin Karliczekc,3,‡, Michael Kupperb,4,§

January 22, 2016

ABSTRACT

The concepts of a conditional set, a conditional inclusion relation and a con-ditional Cartesian product are introduced. The resulting conditional set theoryis sufficiently rich in order to construct a conditional topology, a conditionalreal and functional analysis indicating the possibility ofa mathematical dis-course based on conditional sets. It is proved that the conditional power set isa complete Boolean algebra, and a conditional version of theaxiom of choice,the ultrafilter lemma, Tychonoff’s theorem, the Borel-Lebesgue theorem, theHahn-Banach theorem, the Banach-Alaoglu theorem and the Krein-Šmuliantheorem are shown.

KEYWORDS: Conditional sets, conditional topology, conditional compactness,conditional functional analysis.AMSCLASSIFICATION: 03E70

AUTHORSINFOa CAFR and Department of Mathematics, ShanghaiJiao Tong Universityb Department of Mathematics and Statistics, Univer-sity of Konstanzc Department of Mathematics, Humboldt Universityof Berlin1 drapeau@sjtu.edu.cn2 asgar.jamneshan@uni-konstanz.de3 karliczm@math.hu-berlin.de4 kupper@uni-konstanz.de

∗ Funding: MATHEON project E.11† Funding: Berlin Mathematical School‡ Funding: Konsul Karl und Dr. Gabriele SandmannStiftung§ DFG Project KU 2740/2-1

PAPER INFO

We thank Fares Maalouf, Stephan Müller andMartin Streckfuß for fruitful discussions. We wishto express our gratitude to an anonymous referee fora careful reading of the manuscript and commentswhich essentially improved the presentation.

1 Introduction

Conditional set theory is an approach to study the local or dynamic behavior of structures whose localor dynamic behavior is determined by the information encoded in a measure space, or more generally,in a complete Boolean algebra. By constructing an analysis conditioned on a complete Boolean alge-bra, conditional set theory makes available analytic toolsfor this purpose. In the case of the associatedmeasure algebra, it provides an alternative to measurable selection techniques, and extends the results intopologicalL0-modules obtained in Filipovic et al. [13] and Cheridito et al. [8], initially motivated bydual representations of conditional risk measures [14].

In the following, we briefly introduce conditional set theory. A conditional setX is a collection ofobjectsx|a for x in a non-empty setX anda in a complete Boolean algebraA such that

• a = b wheneverx|a = y|b,

• x|b = y|b impliesx|a = y|a for all a, b ∈ A with a ≤ b, and

• for any partition of unity(ai) in A and a family(xi) of elements inX there exists exactly oneelementx in X such thatx|ai = xi|ai for all i.

In order to introduce a conditional inclusion relation, it is necessary to specify conditional subsets of aconditional setX. A conditional subset ofX is the collection of objectsY|b := y|a : y ∈ Y, a ≤b for someb ∈ A and some non-empty subsetY of X that is stable under pasting of its elements

1

along partitions of unity inA. A conditional subsetY|b is a conditional set on the relative algebraAb.A conditional subsetY|b of X is conditionally included in another conditional subsetZ|c if Y|b ⊆Z|c. It can be shown that the collection of all conditional subsets of X together with the conditionalinclusion relation forms a complete Boolean algebra. The induced operations of conditional intersection,conditional union and conditional complement conserve thestructure of a conditional set, and satisfythe Boolean laws known from naïve set theory. By giving meaning to a conditional Cartesian product,conditional relations and functions can be defined as conditional subsets of the conditional product of twoconditional sets, respectively.

A proof of a conditional version of a classical result is an adaptation of an existing classical proof. Inthis adaptation process, it is helpful to recognize the following principles. The first principle is exhaustionwhich establishes the largest conditiona for which a conditional property is satisfied. The second princi-ple is conditional negation which is stronger than classical negation. Conditional negation negates locallya conditional property on all conditionsa > 0. The third principle is localization. A conditional structureon or a statement about a conditional setX can equivalently be stated on the conditional setX|b for anyb < 1 by passing from the complete Boolean algebraA to its complete relative algebraAb. In particular,the restriction of a true statement aboutX to X|b remains true. The forth principle is bottom-up. Itmakes a relation between a conditional concept on a conditional setX and its classical counterpart on theunderlying setX . For instance, we analyze this relation for the concepts of continuity and convergencein Section3.

Conditional set theory is closely related to the topos of sheaves over a complete Boolean algebra orBoolean-valued models of ZFC, respectively, see Mac Lane and Moerdijk [24] for an introduction tosheaves in logic, see Bell [2] and Kusraev and Kutateladze [23] for an introduction to Boolean-valuedmodels and to Boolean-valued analysis, respectively, and see Jamneshan [20] for the connection of con-ditional sets to sheaves and to Boolean-valued sets. Conditional set theory is an extension of the con-ditional analysis’ results for topologicalL0-modules in [8, 10, 13, 27]. Conditional set operations on(L0)d are introduced in Streckfuß [26]. In this article, it is shown thatL0 is isomorphic to the conditionalreal numbers when the complete Boolean algebra is the measure algebra associated to aσ-finite measurespace. Hence the conditional analysis’ results obtained inL0-theory are recovered in conditional set the-ory, and conditional set theory provides a framework for further studies of stableL0-modules. In Guo[18], conditional separation and duality results for topologicalL0-modules in [13] are related to the re-spective results in randomly normed modules, see Haydon et al. [19] and its references for an introductionto randomly normed spaces. In [11, 12], Eisele and Taieb prove a conditional version of some classicaltheorems from functional analysis for modules overL∞. Recently, a Hahn-Banach theorem for modulesover Stonean algebras has been proved in Cerreia-Vioglio etal. [4]. A conditional version of Mazur’slemma forL0-modules is shown in Zapata-García [27]. Conditional analysis is successfully applied todynamic and conditional risk measures and decision theory in Filipovic et al. [14], Bielecki et al. [3],Frittelli and Maggis [17] and Jamneshan and Drapeau [9], to backward stochastic differential equationsin Cheridito and Hu [5] and Cheridito and Stadje [6], and to optimization problems in equilibrium andprincipal-agent models in Horst et al. [7] and Horst and Backhoff [1].

This paper is organized as follows. Conditional sets, conditional set operations, conditional relations,conditional families, conditional countability and a conditional axiom of choice are introduced in Section2. In Section3, conditional topological spaces and the concepts of conditional continuity, conditionalconvergence and conditional compactness are defined with the aim to prove a conditional version of Ty-chonoff’s theorem. In Section4, the conditional real line is constructed, and conditionalmetric spacesare defined the conditional compactness of which is characterized by a conditional Borel-Lebesgue the-orem. In Section5, conditional topological vector spaces are introduced anda conditional version of theHahn-Banach theorem, the Banach-Alaoglu theorem and the Krein-Šmulian theorem are shown. In thelast two sections, the main theorems are proved. A complete account can be found in Jamneshan [21] and

2

Karliczek [22], respectively.

2 Conditional set theory

LetA = (A,∧,∨, c, 0, 1) be a complete Boolean algebra. Examples are the power set algebra of some set,the measure algebra associated to aσ-finite measure space(Ω,F , µ),1 the quotient algebraB/I whereB is the Borelσ-algebra of a separable metric space andI theσ-ideal of meager sets, see [25, p. 182,Proposition 12.9], and the Boolean algebra of all projective bands of a Riesz space, see [16, Volume 3,p. 232, Theorem 352Q]. Recall thatA together with the relationa ≤ b whenevera ∧ b = a is a completecomplemented distributive lattice. The relative algebra of A with respect to somea ∈ A is denoted byAa := b ∈ A : b ≤ a. For a family(ai) = (ai)i∈I of elements inA, its supremum is denoted by∨ai = ∨i∈Iai and its infimum by∧ai = ∧i∈Iai.2 A partition ofa ∈ A is a family(ai) of elements ofAsuch thatai∧aj = 0 wheneveri 6= j and∨ai = a. Denote byp(a) the set of all partitions ofa. For everyfamily (ai)i∈I of elements inA there exists(bi)i∈I ∈ p(∨ai) such thatbi ≤ ai for all i ∈ I. Indeed, bythe well-ordering theorem there exists a well-ordering on the indexI, and definebi := ai ∧ (∨j<ibj)

c

for eachi ∈ I. Equalities and inequalities between measurable functions are always understood in thealmost sure sense whenever a measure is fixed.

Definition 2.1. A conditional setX of a non-empty setX and a complete Boolean algebraA is a collec-tion of objectsx|a for x ∈ X anda ∈ A such that

(C1) if x|a = y|b, thena = b;3

(C2) if x, y ∈ X anda, b ∈ A with a ≤ b, thenx|b = y|b impliesx|a = y|a;

(C3) if (ai) ∈ p(1) and(xi) is a family of elements inX , then there exists exactly one elementx ∈ X

such thatx|ai = xi|ai for all i.

Condition(C2) is calledconsistencyand(C3) is namedstability. For (ai) ∈ p(1) and a family(xi) ofelements inX , the unique elementx ∈ X such thatx|ai = xi|ai for all i is theconcatenationof thefamily (xi) along the partition(ai), and denoted by

∑

xi|ai. For finite partitions, the concatenation isdenoted byx1|a1 + . . .+ xn|an.

Remark 2.2.Let x, y ∈ X anda ∈ A such thatx|1 = y|a. Then it follows from(C1) thata = 1 andfrom (C3) thatx = y. In particular,X is in bijection withx|1 : x ∈ X. Furthermore, it follows from(C3) thatx|0 = y|0 for everyx, y ∈ X . In particular,x|0 : x ∈ X consists of one element.

Examples 2.3.1) Every conditional set can be identified with the collection of equivalence classes onthe productX ×A for the equivalence relation(x, a) ∼ (y, b) wheneverx|a = y|b.

2) LetA = 0, 1 be the trivial algebra andX a non-empty set. The collectionX of objectsx|1 = x

andx|0 = X × 0 for all x ∈ X , is a conditional set.

3) LetA be a complete Boolean algebra. ThenX = A×A/∼ where(a, b) ∼ (c, d) whenevera∧b = c∧dandb = d, is a conditional set with equivalence classesa|b.

1The associated measure algebra is the quotient Boolean algebra ofF by theσ-ideal ofµ-null sets, see [25, p. 233, Example14.27]

2As usual, we apply the conventions∨∅ = 0 and∧∅ = 1.3In the first version of this paper, the assumption "identity"was required. This condition was dropped during the revision process,

and we are grateful to José Miguel Zapata García for pointingout to us the lack of this condition in the current version andsuggesting the appropriate replacement.

3

4) Let (Ω,F , µ) be aσ-finite measure space,A the associated measure algebra andL0 the set of allequivalence classes of measurable functionsX : Ω → R which coincideµ-almost everywhere. Denotethe equivalence classes inF by a = [A] and the equivalence classes inL0 by x = [X ]. The collectionL0 of objectsx|a = y ∈ L0 : Y 1A = X1A is a conditional set.

5) Conditional set of step functions: LetA be a complete Boolean algebra andE a non-empty set. Weconsider the collection of all families(xi, ai) of elements inE×A where(ai) ∈ p(1). On this collectionwe define the equivalence relation(xi, ai) ∼ (yj , bj) if ∨ai : xi = z = ∨bj : yj = z for all z ∈ E,and we denote byX the respective set of equivalence classes[xi, ai]. Inspection shows that we can makeX into a conditional setX by considering the collection of objects

[xi, ai]|a := [yj , bj ] ∈ X : ∨ ai : xi = z ∧ a = ∨bj : yj = z ∧ a for all z ∈ E .

This construction can be seen as the conditional set of step functions onA with values inE. Indeed,each[x, 1] can be uniquely identified withx ∈ E. Since by stability, elements inX can be writtenas

∑

[xi, 1]|ai it justifies the notation∑

xi|ai for the elements ofX that can be interpreted as the stepfunction taking the valuexi ∈ E onai.

In case thatE is eitherN or Q we denote the respective conditional set of step functions by N or Q,and call them theconditional natural numbersand conditional rational numbers, respectively. Thecorresponding generating sets are denoted byN =

∑

ni|ai : (ai) ∈ p(1), (ni) is a family inN andQ =

∑

qi|ai : (ai) ∈ p(1), (qi) is a family inQ. ♦

Proposition 2.4. LetX be a conditional set.

(i) For all (ai), (bj) ∈ p(1) and families(xij) of elements inX , it holds

∑

j

(

∑

i

xij |ai)

|bj =∑

i,j

xij |ai ∧ bj .

(ii) For all b ∈ A, (ai) ∈ p(b) and families(xi) of elements inX , there existsx ∈ X such thatx|ai = xi|ai for all i, and ify ∈ X is such thaty|ai = xi|ai for all i, thenx|b = y|b.4

Proof. (i) Denote byyj :=∑

xij |ai for eachj andy :=∑

yj |bj . Sincey|bj = yj |bj, one hasy|ai ∧bj = yj |ai ∧ bj by consistency. Similarly, it follows fromyj |ai = xij |ai thatyj |ai ∧ bj = xij |ai ∧ bj .Hence,y|ai ∧ bj = xij |ai ∧ bj for all i, j, and thusy =

∑

xij |ai ∧ bj .

(ii) Let x, y ∈ X be such thatx|ai = xi|ai for all i andx|bc = w|bc, andy|ai = xi|ai for all i andy|bc = z|bc for somew, z ∈ X . Let v := y|b + w|bc. Sincev|b = y|b, it holdsv|ai = y|ai for alli due to consistency. Sincey|ai = xi|ai = x|ai for all i, one hasv = x by stability. By consistency,x|b = v|b = y|b.

Definition 2.5. Let X be a conditional set. A subsetY of X is calledstableif it is non-empty and

Y =

∑

yi|ai : (ai) ∈ p(1), (yi) is a family of elements inY

.

Denote byS(X) the set of allY ⊆ X which are stable.

4Whenever there is no risk of confusion, we denote byx|b =∑

xi|ai.

4

For every non-emptyY ⊆ X , the collection of those∑

yi|ai, where(ai) ∈ p(1) and(yi) is a familyof elements inY , is the smallest stable set containingY due to Proposition2.4, and is referred to as thestable hulls(Y ) of Y .

Examples 2.6.1) For a conditional setX of a non-empty setX and the trivial algebraA = 0, 1, anynon-empty subset ofX is stable.

2) For any conditional setX, every singletonx, x ∈ X , is stable.

3) For the conditional setL0, the set[x, y] := z ∈ L0 : X ≤ Z ≤ Y is stable for anyx, y ∈ L0 withX ≤ Y . Further examples of stable subsets ofL0 can be found in [4, 8, 10, 13]. ♦

By stability, everyY ∈ S(X) generates a conditional setY := y|a : y ∈ Y, a ∈ A.

Definition 2.7. LetX be a conditional set. LetP (X) be the collection of all conditional setsY generatedby Y ∈ S(X) and define

P(X) := Y|a = y|b : y ∈ Y, b ≤ a : Y ∈ P (X), a ∈ A ,

which we call theconditional power setof X.

Inspection shows thatP(X) is a conditional set ofP (X). Note that every elementY|a in the conditionalpower set ofX is itself a conditional set ofY |a := y|a : y ∈ Y and the relative algebraAa, for theconditioning(y|a)|b := y|b, b ≤ a.

Definition 2.8. Let X andY be two conditional sets ofX,A andY,B respectively. We say thatY isconditionally includedin X and writeY ⊑ X if B = Aa for somea ∈ A andY = Z|a for someZ ∈ P (X). We say thatY is aconditional subsetof X ona.

Depending on the context, we writeY orY|a for a conditional subset ofX ona. Note that

P(X) = Y ⊑ X : Y conditional set .

By inspection,⊑ is a partial order onP(X) with greatest elementX = X|1 and least elementX|0. Everysingletonx, x ∈ X , is stable and defines a conditional setx := x|a : a ∈ A called aconditionalelement ofX.5 Denote bys(Y ) ∈ P (X) the conditional set generated by the stable hull of some non-empty subsetY ⊆ X .

Theorem 2.9. Let X be a conditional set. Then(P(X),⊑) is a complete complemented distributivelattice.

Proof. First we prove completeness, second complementation and third distributivity.

Step 1 : Let (Yi|ai) be a non-empty family of conditional subsets ofX. First we construct the supremumof (Yi|ai) and second its infimum. Fixz ∈ X , and let

Y =

∑

yi|bi + z| ∧ aci : (bi) ∈ p(∨ai) with bi ≤ ai, yi ∈ Yi for eachi

where we used the well-ordering theorem to find a partition(bi) ∈ p(∨ai) with bi ≤ ai for all i, andProposition2.4 to construct each of the concatenations

∑

yi|bi + z| ∧ aci . We want to show thatY is

5Although the conditional elementsx of X are formally sets, they are atoms among the conditional subsets ofX on1, and are inone-to-one relation withx ∈ X. In the case thatY is a conditional set ona, a conditional elementy of Y is ona.

5

stable. To this end, let(cj) ∈ p(1) and(xj) be a family of elements inY wherexj =∑

yij |bij + z| ∧ acifor eachj. For eachi, definebi = ∨bij ∧ cj andyi =

∑

yij |cj . Inspection shows that(bi) ∈ p(∨ai)with bi ≤ ai. By stability, one hasyi ∈ Yi for eachi. From Proposition2.4 it follows that

∑

xj |cj =∑

yi|bi + z| ∧ aci ∈ Y.

With Y being the conditional set generated by the stable setY , we show that

⊔Yi|ai := Y| ∨ ai (2.1)

is the supremum of(Yi|ai). Indeed, for anyi, yi ∈ Yi and some arbitraryy ∈ Y , it holdsw :=

yi|ai + y|aci ∈ Y due to Proposition2.4. Sincew|ai = yi|ai, it follows thatY| ∨ ai is an upper bounddue to consistency. For any other upper boundW|c, it must holdai ≤ c for eachi, and therefore∨ai ≤ c.Moreover, for alli and everyyi ∈ Yi there isw ∈ W such thatyi|ai = w|ai. By Proposition2.4, for ally ∈ Y there isw ∈ W with y| ∨ ai = w| ∨ ai. By consistency,Y| ∨ ai is the least upper bound.

We want to show that there exists an infimum of(Yi|ai). Let

M = a : a ≤ ∧ai, there existsx ∈ X such that for alli there isyi ∈ Yi with x|a = yi|a,

andb = ∨M . We show thatb is attained. Let(cj) be a family of elements inM with ∨cj = b, that is foreachj there existsxj ∈ X and for alli there existsyij ∈ Yi with xj |cj = yij |cj . By the well-orderingtheorem, there is(dj) ∈ p(b) such thatdj ≤ cj for all j. By consistency,dj ∈ M for eachj. ByProposition2.4, one has

∑

xj |dj =∑

yij |dj for all i, and thusb ∈ M . Next we show that

Y := x ∈ X : for all i there existsyi ∈ Yi with x|b = yi|b

is stable. To this end, let(cj) ∈ p(1) and(xj) be a family of elements inY . For allj and everyi there isyij ∈ Yi with xj |b = yij |b. Setyi =

∑

yij |cj for eachi. By Proposition2.4, it holdsyi|b = xj |b, andthus(

∑

xj |cj)|b = yi|b. By construction,

⊓Yi|ai := Y|b (2.2)

is a lower bound of(Yi|ai). For any other lower boundW|c, it holdsc ≤ ai for all i. Moreover, for allw ∈ W and everyi there existsyi ∈ Yi such thatw|c = yi|c. Thereforec ≤ b. By consistency, it holdsw|d : w ∈ W,d ≤ c ⊆ y|d : y ∈ Y, d ≤ c. Thus⊓Yi|ai is the greatest lower bound of(Yi|ai).

Step 2 : We want to show that for all conditional subsetsY|a of X it holds

Y|a ⊓ (Y|a)⊏ = X|0 and Y|a ⊔ (Y|a)⊏ = X|1,

where(Y|a)⊏ := ⊔Z|c ∈ P(X) : Z|c ⊓Y|a = X|0 (2.3)

is the complement. By completeness which has been proved inStep 1, it holds(Y|a)⊏ ∈ P(X). Suppose(Y|a)⊏ is of the formW|b for someW ∈ P (X) andb ∈ A.

As for the first statement, by way of contradiction, we may assume thatY|a ⊓ W|b = Z|c for somec > 0. Thus there existsy ∈ Y andw ∈ W such thaty|c = w|c. However, this impliesy|c satisfiesy|c ⊓Y|a 6= X|0 which is contradictory. Hencec = 0, and thusY|a ⊓W|b = X|0.

As for the second statement, we may assume thatY|a 6= X|1, since otherwiseY|a⊔W|b = X|1⊔X|0 =

X|1. SupposeY|a ⊔ W|b = Z|a ∨ b for someZ ∈ P (X), and letx ∈ X . Thenx ⊓ Y|a = y|c for

6

somey ∈ Y andc = ∨a′ : a′ ≤ a, there existsy ∈ Y with x|a′ = y|a′ andx ⊓W|b = w|d for somew ∈ W andd = ∨b′ : b′ ≤ b, there existsw ∈ W with x|b′ = w|b′. SinceY|a⊓W|b = X|0, it musthold c ∧ d = 0 andc ∨ d = 1.6 It follows from the construction of the supremum⊔ thata ∨ b = 1 andx = x|c+ x|d ∈ Z. ThusZ|a ∨ b = X|1.

Step 3 : Let Yk|ak ∈ P(X) for k = 1, 2, 3. Since it has already been shown that(P(X),≤) is a lattice,both distributive laws are equivalent, see [25, p. 15, Lemma 1.17]. It remains to prove that

(Y1|a1 ⊓Y2|a2) ⊔ (Y1|a1 ⊓Y3|a3) = Y1|a1 ⊓ (Y2|a2 ⊔Y3|a3).

On the one hand, in every lattice it holds

(Y1|a1 ⊓Y2|a2) ⊔ (Y1|a1 ⊓Y3|a3) ≤ Y1|a1 ⊓ (Y2|a2 ⊔Y3|a3).

On the other hand, suppose(Y1|a1 ⊓ Y2|a2) ⊔ (Y1|a1 ⊓ Y3|a3) = V|c for someV ∈ P (X) andc ∈ A. Without loss of generality, we may assume thatY1|a1 ⊓ (Y2|a2 ⊔ Y3|a3) is of the formW|1 for someW ∈ P (X). By the construction of the infimum⊓, this immediately impliesa1 = 1 andY2|a2⊔Y3|a3 = Z|1 for someZ ∈ P (X). Moreover, for everyw ∈ W there arey ∈ Y1 andz ∈ Z suchthatw|1 = y|1 = z|1. By the construction of the supremum⊔, there exists(b, bc) ∈ p(1) with b ≤ a2andbc ≤ a3, andv ∈ Y2 andu ∈ Y3 such thatz = v|b + u|bc. By consistency,w|b = y|b = z|b = v|bandw|bc = y|bc = z|bc = u|bc. Sincew = w|b + w|bc, it follows w ∈ V andc = 1. ThusW|1 ⊑ V|1which finishes the proof.

The operations⊔,⊓ and⊏, given in (2.1), (2.2) and (2.3), are namedconditional union, intersectionandcomplement, respectively. As a consequence of standard results on complete complemented distributivelattices and Boolean algebras, see [25, p. 14], we have:

Corollary 2.10. For every conditional setX,

P(X) = (P(X),⊔,⊓,⊏,X|0,X)

is a complete Boolean algebra.

Remark 2.11.The complete Boolean algebraP(X) is atomic if, and only if,A is atomic. Indeed, letAbe the set of atoms ofA. Then the set of atoms ofP(X) is x|a : a ∈ A, x ∈ X. Conversely, ifA isatomless then for eacha > 0 there exists0 < b < a such thatx|b ⊑ x|a andx|b 6= x|a for all x ∈ X .Analogously, one can verify that the distributivity law ofP(X) coincides with the distributive law ofA.

Corollary 2.12. LetX be a conditional set.

(i) De Morgan’s law: For any non-empty family(Yi) of conditional subsets ofX, it holds

(⊔Yi)⊏ = ⊓(Yi)

⊏.

(ii) Distributivity: For any non-empty family(Yij)i∈I, j∈J of conditional subsets ofX whereJ isarbitrary andI is finite, it holds

⊓i∈I ⊔j∈J Yij = ⊔

⊓i∈IYif(i) : f ∈ JI

.

6By inspection, the negation of either of the two conditions leads immediately to a contradiction to the construction of the infimum⊓ and the complement⊏.

7

(iii) Associativity: For any non-empty family(Yij)i∈I, j∈J of conditional subsets ofX whereI, J arearbitrary, it holds

⊔i∈I(⊔j∈JYij) = ⊔i∈I, j∈JYij .

Proof. All three properties are satisfied in every complete Booleanalgebra, see [25, p. 22, Lemma 1.33].

Lemma 2.13. LetX be a conditional set.

(i) For Y1,Y2 ⊑ X such thatY1 ⊓Y2 is on1, one hasY1 ∩ Y2 6= ∅.

(ii) For Y1,Y2 ⊑ X on1 such thatY1 ⊑ Y2, one hasY1 ⊆ Y2 andY1 ⊆ Y2.

(iii) Let (Yi) be a stable collection of conditional subsets ofX each on1. Then one has⊔Yi = W

whereW = ∪Yi.

Proof. The first two statements are immediate from the definitions. As for the third one, let

W =

∑

yj |aj : (aj) ∈ p(1), yj ∈ Yij for someij

.

It follows that∪Yi ⊆ W . Conversely, lety =∑

yj|aj ∈ W , so thaty ∈∑

Yij |aj . By stability,∑

Yij |aj = Yik for someik and thereforey ∈ ∪Yi showing thatW = ∪Yi. By definition of theconditional union, it holds⊔Yi = W.

2.1 Conditional relation and function

Definition 2.14. Let I be a non-empty index set andXi be a conditional set for eachi ∈ I. The collectionof objects(xi)i∈I |a := (xi|a)i∈I for (xi)i∈I ∈

∏

i∈I Xi anda ∈ A is called theconditional productoftheXi, and is denoted by

∏

i∈I Xi.

Note that the conditional product is a conditional set. For aconditional setX and a non-empty index setI, we writeXI :=

∏

i∈I X andXn :=∏

1≤k≤n X for anyn ∈ N.

Definition 2.15. Let X andY be conditional sets. Aconditional binary relationonX ×Y is a condi-tional subsetT of X×Y on1.7 We say thatx|a andy|a for two conditional elementsx,y of X×Y are inrelation if(x|a, y|a) ∈ T and writex|aTy|a. A conditional binary relationT onX×X is conditionallyantisymmetric, reflexive, symmetricor transitivewheneverT is antisymmetric, reflexive, symmetric ortransitive in the classical sense. Aconditional partial order6 is a conditionally antisymmetric, reflexiveand transitive binary relation. Given a conditional partial order6, we definex|a < y|a if x|a 6 y|a andx|b 6= y|b for every0 < b ≤ a.8 A conditional partial order6 is conditionally totalif for everyx, y ∈ X

there exists(a, b, c) ∈ p(1) such thatx|a < y|a, y|b < x|b andx|c = y|c. A conditional equivalencerelation is a conditionally symmetric, reflexive and transitive binary relation.

Conditional extrema of a conditionally partially ordered set are defined as classical extrema. For instance,a conditional direction is a conditionally reflexive and transitive binary relation with the property thatevery pairx, y ∈ X has an upper bound.

Examples 2.16.1) OnQ define∑

pi|ai ≤∑

qj |bj wheneverpi ≤ qj for all i, j with ai ∧ bj > 0. It isimmediate from the definition, that the previous binary relation is a stable subset ofQ ×Q. The relatedconditional relation6 on the conditional rational numbersQ is a conditionally total partial order.

7A conditional binary relationT is a classical relationT ⊆ X × Y such that∑

(xi, yi)|ai = (∑

xi|ai,∑

yi|ai) ∈ T for all(ai) ∈ p(1) and every family(xi, yi) of elements inT .

8By consistency, ifx|a 6 y|a it holdsx|b 6 y|b for everyb ≤ a.

8

2) OnL0 definex ≤ y wheneverX ≤ Y . By inspection, this binary relation is a stable subset ofL0×L0.The related conditional relation6 onL0 is a conditionally total partial order. ♦

Definition 2.17. LetX andY be conditional sets. A functionf : X → Y is stableif

f(

∑

xi|ai)

=∑

f(xi)|ai, for all (ai) ∈ p(1) and every family(xi) of elements inX.

A conditional subsetGf of X×Y on1 is thegraph of a conditional functionf : X → Y wheneverGf

is the graph of a stable functionf : X → Y .ForU|b ⊑ X, theconditional imagef(U|b) is defined asZ|b ⊑ Y whereZ = f(x) : x ∈ U. For

V|c ⊑ Y, theconditional preimagef−1(V|c) is defined asW|d ⊑ X where9

d = ∨a : a ≤ c, there arex ∈ X, y ∈ V such thatf(x)|a = y|a

W = x ∈ X : there isy ∈ V with f(x)|d = y|d.

A conditional functionf : X → Y is conditionally injectiveif x, x′ ∈ X with x|a 6= x′|a for alla 6= 0 impliesf(x)|a 6= f(x′)|a for all a 6= 0; it is conditionally surjectiveif f is surjective; and it isconditionally bijectiveif it is conditionally injective and surjective.

Examples 2.18.1) LetX andY be non-empty sets andf : X → Y an injective function. LetX, YandGf be the conditional sets generated byX,Y and the graphGf of f .10 Then the conditional functionf : X → Y is conditionally injective.

2) Let∏

Xi be the conditional product of a family of conditional sets. Thej-th projectionprj :∏

Xi →Xj is a stable function, andprj :

∏

Xi → Xj is called thej-th conditional projection.

3) LetX be a conditional set andY a conditional subset ofX on 1. The embeddingY → X is stable,and the conditional functionY → X is called aconditional embedding.

4) We call the conditional function|·| : Q → Q+ = q : 0 6 q generated byq =∑

qi|ai 7→∑

|qi| |aionQ, theconditional absolute value. ♦

The assertions in the following proposition follow from Theorem2.9.

Proposition 2.19. LetX andY be two conditional sets andf : X → Y a conditional function. For anon-empty family(Ui) of conditional subsets ofX, a non-empty family(Vj) of conditional subsets ofY,conditional subsetsZ1,Z2 of X with Z1 ⊑ Z2 and conditional subsetsW1, W2 of Y with W1 ⊑ W2,andZ ⊑ X andW ⊑ Y, it holds

f(Z1) ⊑ f(Z2), f−1(W1) ⊑ f−1(W2), (2.4)

f(⊔Ui) = ⊔f(Ui), f−1(⊔Vi) = ⊔f−1(Vi), (2.5)

f(⊓Ui) ⊑ ⊓f(Ui), f−1(⊓Vi) = ⊓f−1(Vi), (2.6)

f(Z)⊏ ⊓ f(X) ⊑ f(Z⊏), f−1(W⊏) = f−1(W)⊏, (2.7)

Z ⊑ f−1(f(Z)), f(f−1(W)) ⊑ W. (2.8)

It holds equality on the left-hand side of(2.8) if f is conditionally injective, and on its right-hand side ifW ⊑ f(X).

9The stability ofW follows from the stability off . Note thatd = c wheneverf(X|c) = Y|c.10In the sense of Example2.35).

9

2.2 Conditional family and axiom of choice

Definition 2.20. Let X andI be conditional sets. Astable family(xi) of elements inX is the graphGf = (f(i), i) : i ∈ I wheref : I → X is a conditional function. In particular,

∑

xij |aj = x∑ij |aj

for every(aj) ∈ p(1) and every family(ij) of elements inI. A conditional family(xi) of conditionalelements ofX is the conditional graphGf .

A conditional family (xi) is a conditional element of∏

i∈I Xi whereXi = X for eachi ∈ I. ByProposition2.4, the collection of all stable families(xi) is a stable subset of

∏

i∈I Xi, and we denoteby XI the conditional set corresponding to it. For a conditional natural numbern, we writeXn forX16l6n. Note that

1 6 l 6 n = s(

∑

li|ai : 1 ≤ li ≤ ni, for eachi)

wheren =∑

ni|ai ∈ N . In particular, stability impliesXn =∑

Xni |ai.

Example 2.21.Let X be a conditional set and(I,6) be a conditional direction. A conditional family(xi) of conditional elements ofX is called aconditional net. In case that(I,6) equals to(N,6), aconditional family(xn) is called aconditional sequenceof conditional elements ofX. ♦

Lemma 2.22. Let X be a conditional subset ofN on 1. Suppose thatx 6 k for every conditionalelementx of X and for some conditional elementk of N. Then there exists a conditional bijectionf : X → 1 6 l 6 n for a unique conditional elementn ofN.

Proof. Suppose thatk =∑

ki|bi. For eachi, let Ji be the set of non-empty subsets of1, . . . , ki.For eachMi ∈ Ji, defineaMi

= ∨a : a ≤ bi andn ∈ N : x|a = n|a for somex ∈ X = Mi.Then one has(aMi

) ∈ p(bi) for eachi. Definen =∑

i,Mi∈Jicard(Mi)|aMi

where card(Mi) denotesthe cardinality ofMi. Choosex =

∑

mj |dj ∈ X . On everyaMi∧ dj > 0 it holdsmj ∈ Mi, the

position11 of which in the ordered setMi is denoted bymj,Mi. Definef(x) =

∑

mj,Mi|ai,Mi

∧ dj .Thenf : X → 1 6 l 6 n is stable by Proposition2.4. By construction,n ∈ N is unique andf : X → 1 6 l 6 n a conditional bijection.

Definition 2.23. A conditional setX is conditionally countableif there exists a conditional injectionf : X → N. It is conditionally finiteif there exists a conditional bijectionf : X → 1 6 l 6 n forsome conditional elementn of N.

Example 2.24.The conditional rational numbersQ are conditionally countable since every injectionf : Q → N generates a conditional injectionf : Q → N by Example2.181). ♦

Proposition 2.25. LetX be a conditional set and(Yn) be a conditional sequence of conditional subsetsofX. Then it holds:12

(i)⊔16l6nYl =

∑

(

⊔ni

li=1Yli

)

|ai and ⊓16l6n Yl =∑

(

⊓ni

li=1Ylm

)

|ai ∧ bi

wheren =∑

ni|ai and⊓ni

li=1Yli is a conditional subset onbi for eachi.

(ii) If Yl is conditionally finite for each1 6 l 6 n, then⊔16l6nYl is conditionally finite.

11EachMi is an ordered set of the formnMi1

, . . . , nMikMi

andmj,Miis the index such thatmj = n

Mimj,Mi

.12⊔16l6n and⊓16l6n are understood as the conditional union and intersection over all conditional elementsl such that1 6 l 6

n.

10

(iii) If Yn is conditionally countable for eachn, then⊔Yn is conditionally countable.

Proof. The statements are implied by Proposition2.4.

We close this section by a conditional axiom of choice.

Theorem 2.26. Let X be a conditional set and(Yi) be a conditional family of conditional subsets ofX. Then there exists a conditional family(yi) of conditional elements ofX such thatyi is a conditionalelement ofYi for eachi.

Proof. LetH := (yj)j∈J : yj ∈ Yj for eachj ∈ J, J ∈ S(I) .

The setH is non-empty since it includes every one-element family. Define an ordering onH by

(yj)j∈J ≤ (yj)j∈J wheneverJ ⊆ J andyj = yj for all j ∈ J.

Let (yj)j∈Jαbe a chain inH , and put

J =

∑

jβ |aβ : (aβ) ∈ p(1), jβ ∈ Jαβfor eachβ

.

By Proposition2.4, one hasJ ∈ S(I). For j =∑

jβ |aβ ∈ J , defineyj =∑

yjβ |aβ. Since(Yi) isa conditional family, it holds(yj)j∈J ∈ H . Inspection shows that(yj)j∈Jα

≤ (yj)j∈J for eachα. ByZorn’s lemma, there exists a maximal element(yj)j∈J∗ in H . By way of contradiction, suppose thereexistsi0 ∈ I such thatyj ⊓Yi0 on somebj < 1 for all j ∈ J∗. Let J = j|a+ i0|a

c : a ∈ A, j ∈ J∗,pick someyi0 ∈ Yi0 and defineyj = yj |a + yi0 |a

c for eachj ∈ J . Then(yj)j∈J is an element inH .However, one has(yj)j∈J∗ < (yj)j∈J which is the desired contradiction.

3 Conditional topology

Let X be a conditional set. From the construction of the conditional power set it follows that

• P(P(X)) is a conditional set ofP (P(X));

• P (P(X)) are the conditional subsets ofP(X) on1;

• elements inP (P(X)) are generated byS(P(X));

• S(P(X)) are stable subsets ofP (X).

Hence, an element inP(P(X)) is a conditional collection of conditional subsets ofX and an elementin S(P(X)) is a stable collection of conditional subsets ofX on1. A stable collectionB of conditionalsubsets ofX on 1 is in one-to-one relation to a stable collectionB of stable subsets ofX , the relationbeing given byB = Y : Y ∈ B andB = Y : Y ∈ B.

Definition 3.1. Let X be a conditional set andT a conditional collection of conditional subsets ofX. Tis called aconditional topologyonX whenever

(i) X ∈ T ,

(ii) if O1,O2 ∈ T , thenO1 ⊓O2 ∈ T ,

(iii) if (Oi) is a non-empty collection inT , then⊔Oi ∈ T .

11

The pair(X, T ) is called aconditional topological space. A conditional setO ∈ T is calledconditionallyopen. A conditional subsetF of X is calledconditionally closedwheneverF⊏ ∈ T .13 Given twoconditional topologiesT1 andT2, T1 is said to beconditionally weakerthanT2 wheneverT1 ⊑ T2. Aconditional collectionB of conditional subsets ofX is aconditional topological basewhenever

(i) ⊔B = X,

(ii) if O1,O2 ∈ B andx is a conditional element ofO1 ⊓O2, then there existsO3 ∈ B such thatx isa conditional element ofO3 andO3 ⊑ O1 ⊓O2.

The conditional topologyconditionally generatedby a conditional collectionG of conditional subsets ofX is

T G := ⊓T : T conditional topology,G ⊑ T .

For a conditional topological baseB, inspection shows

T B = ⊔Oi : (Oi) non-empty collection inB.

Example 3.2. For conditional elementsq, r of Q such thatr > 0, define the conditional set14

Br(q) := p : |q− p| < r .

The conditional collectionB of conditional sets generated by the stable collection

Br(q) : q, r of Q with r > 0

is a conditional topological base ofT B called theconditional Euclidean topologyonQ. ♦

Definition 3.3. Given a conditional topological space(X, T ) and a conditional subsetY of X, thecon-ditional interior of Y is defined by

int(Y) := ⊔O : O conditionally open,O ⊑ Y ,

and itsconditional closureby

cl(Y) := ⊓F : F conditionally closed,Y ⊑ F.

By the duality principle, one has

cl(Y)⊏ = int(

Y⊏)

and int(Y)⊏ = cl(

Y⊏)

.

13 Due to the duality principle in Boolean algebras, see [25, p. 13], the conditional collection of all conditionally closed sets satisfiesthe dual properties of the conditional collection of all conditionally open sets.

14See Example2.184) for the definition of the conditional absolute value.

12

Definition 3.4. Let (X, T ) be a conditional topological space andx a conditional element ofX. Aconditional subsetU of X is aconditional neighborhoodof x, if there exists a conditionally open setO

such thatx is a conditional element ofO andO ⊑ U.15 Let

U(x) = U : U conditional neighborhood ofx

denote the stable collection of all conditional neighborhoods ofx. A conditional neighborhood baseofx is a stable collectionV of conditional subsets ofX on 1 such that for every conditional neighborhoodU of x there existsV ∈ V such thatx is a conditional element ofV andV ⊑ U.

A conditional topological space(X, T ) is conditionally first countableif every conditional elementxof X has a conditionally countable neighborhood base. It isconditionally second countableif T isconditionally generated by a conditionally countable topological base. It isconditionally Hausdorffif forevery pairx,y of conditional elements ofX with x ⊓ y = X|0 there exists a conditional neighborhoodU of x and a conditional neighborhoodV of y such thatU ⊓ V = X|0. A conditional subsetY ofX is conditionally denseif cl(Y) = X, and(X, T ) is conditionally separableif X has a conditionallycountable dense subset.

Let B be a classical topological base onX . We denote byT B the classical topology generated byB. Furthermore, denote by cl(Y ) the closure and by int(Y ) the interior of someY ⊆ X with respect toT B.

Proposition 3.5. Let X be a conditional set,B a stable collection of stable subsets ofX andB thecorresponding conditional collection of conditional subsets ofX. ThenB is a conditional topologicalbase onX if, and only if,B is a classical topological base onX . Moreover, it holds

O ∈ TB : O ∈ S(X) = O ∈ S(X) : O ∈ T B.

Proof. First assume thatB is a conditional topological base. By Lemma2.13, one has∪B = X . LetO1, O2 ∈ B andx ∈ O1 ∩ O2. By the definition of conditional intersection,x is a conditional elementof O1 ⊓O2. Hence there existsO3 ∈ B such thatx is a conditional element ofO3 andO3 ⊑ O1 ⊓O2.By Lemma2.13, one concludesx ∈ O3 andO3 ⊆ O1 ∩ O2. Second assume thatB is a classical base.By Lemma2.13, it holds⊔B = X. LetO1,O2 ∈ B, supposeO1 ⊓O2 is a conditional subset ofX onb,and letx|b be a conditional element ofO1 ⊓O2. Since∪B = X , there existsO ∈ B such thatx ∈ O.SinceB is stable, it holdsO1|b+O|bc, O2|b+O|bc ∈ B. Moreover,x ∈ (O1|b+O|bc)∩ (O2|b+O|bc).Hence there existsO3 ∈ B such thatx ∈ O3 andO3 ⊆ (O1|b + O|bc) ∩ (O2|b + O|bc). By (C2) andLemma2.13, x|b is a conditional element ofO3|b andO3|b ⊑ O1 ⊓O2.

Example 3.6. LetL0++ := x ∈ L0 : X > 0. Then for everyx ∈ L0 and eachr ∈ L0

++,

Br(x) :=

y ∈ L0 : |X − Y | < R

is a stable subset ofL0. The stable collection

B :=

Br(x) : x ∈ L0, r ∈ L0++

generates theL0-topology introduced in [13]. According to Proposition3.5, the corresponding condi-tional collectionB of conditional subsets ofL0 is a conditional topological base generatingT B which isconditionally Hausdorff and separable, see [21, Lemma 5.3.2]. ♦

15In particular,U is on1.

13

Proposition 3.7. Let (X, T ) be a conditional topological space andY a conditional subset ofX on 1.Then it holds

int(Y) = ⊔x|b : U|b ⊑ Y for some conditional neighborhoodU of x ,

cl(Y) = ⊔x : U ⊓Y is on1 for every conditional neighborhoodU of x

whereb = ∨a : O ⊑ Y for some conditional open setO ona.

Proof. The first assertion is immediate from the definitions. As for the second one, assume thatx isa conditional element ofF for all conditionally closed setsF with Y ⊑ F. Suppose, for the sake ofcontradiction, that there exists a conditional open neighborhoodO of x such thatO ⊓ Y is ona < 1.Thenx|ac ⊑ Y⊏|ac. SinceO⊏|ac is conditionally closed,O⊏|ac + X|a is conditionally closed withY ⊑ O⊏|ac +X|a, which is the desired contradiction. Conversely, assume thatx is such thatU ⊓Y ison1 for every conditional neighborhoodU of x. Suppose, for the sake of contradiction, that there existsa conditionally closed setF with Y ⊑ F such thatx ⊓ F = x|c for somec < 1. Thenx|cc ⊑ F⊏|cc.SinceF⊏|cc is a conditionally open neighborhood ofx|cc, it follows thatF⊏|cc +X|c is a conditionallyopen neighborhood ofx. By assumption,(F⊏|cc+X|c)⊓Y is on1. However, this contradictsY ⊑ F.

3.1 Conditional continuity

Definition 3.8. Let (X, T ) and(X′, T ′) be conditional topological spaces. A conditional functionf :

X → X′ is said to beconditionally continuousat the conditional elementx of X if f−1(U) is a condi-tional neighborhood ofx for all conditional neighborhoodsU of f(x). If f is conditionally continuousat every conditional elementx of X, thenf is said to be conditionally continuous. Let(Xi, Ti)i∈I bea non-empty family of conditional topological spaces and(fi)i∈I be a family of conditional functionsfi : X → Xi. Theconditional initial topologyon X for the family (fi)i∈I is the conditional topologygenerated bys(G) whereG := f−1

i (Oi) : f−1i (Oi) on1,Oi ∈ Ti, i ∈ I.16

Examples 3.9.1) Let (X, T ) be a conditional topological space,Y a conditional subset ofX on1 andf : Y → X a conditional embedding, see Example2.183). Theconditional relative topologyof T withrespect toY is the conditional initial topology forf .

2) Let(Xi, Ti) be a non-empty family ofconditional topological spaces. The conditional product topol-ogy on

∏

Xi is the conditional initial topology for the family of conditional projections(pri). ♦

Proposition 3.10. Let (X, T ) and(X′, T ′) be conditional topological spaces andf : X → X′ a condi-tional function. The following are equivalent:

(i) The conditional functionf is conditionally continuous.

(ii) f−1(O′) is conditionally open for every conditionally open setO′.

(iii) f−1(F′) is conditionally closed for every conditionally closed setF′.

Proof. The assertions are immediate from the definitions.

Proposition 3.11. Let X and X′ be two conditional sets,B and B′ stable collections of stable setswhich are a base of a topology onX andX ′, respectively, andB andB′ the corresponding conditionaltopological bases onX and X′, respectively. A conditional functionf : X → X′ is conditionallycontinuous if, and only if,f : X → X ′ is continuous with respect to the topologiesT B andT B

′

.

16Note that iff−1

i (Oi) is ona < 1 for someOi ∈ Ti, thenf−1

i (Oi|a+Xi|ac) is on1.

14

Proof. Assume thatf : X → X ′ is continuous. LetO ∈ B′, and suppose thatf−1(O) is a conditionalsubset ofX ona. By Proposition3.5, one has

f−1(O)|a +X |ac = f−1(O|a+X ′|ac) ∈ O ∈ TB : O ∈ S(X) = O ∈ S(X) : O ∈ T B.

Thusf−1(O) = f−1(O|a + X′|ac)|a ∈ T B. Conversely, assume thatf : X → X′ is conditionallycontinuous and letO ∈ B′. Without loss of generality, we may assume thatf−1(O) 6= ∅. By Proposition3.5, one hasf−1(O) ∈ O ∈ S(X) : O ∈ T B = O ∈ T B : O ∈ S(X), and thereforef−1(O) ∈T B.

3.2 Conditional filters

Definition 3.12. Let X be a conditional set. Aconditional filterF onX is a stable collection of condi-tional subsets ofX on1 satisfying the following conditions:

(i) if Y ∈ F andY ⊑ Z ⊑ X, thenZ ∈ F ;

(ii) if Y,Z ∈ F , thenY ⊓ Z ∈ F .

A conditional filterF is conditionally finerthan a conditional filterF ′ if F ′ ⊑ F .17 A conditionalultrafilter is a maximal element in the set of all conditional filters onX. A stable collectionB of condi-tional subsets ofX on 1 is aconditional filter baseif for everyY1,Y2 ∈ B there existsY3 ∈ B withY3 ⊑ Y1 ⊓Y2.

Remark 3.13.By consistency, ifF is a conditional filter onX, thenF|a is a conditional filter onX|a.Moreover, letF := Yi|ai : i be a collection of conditional subsets ofX not necessarily on1 such that∑

(

Yij |aij)

|bj =∑

Yij |aij ∧ bj ∈ F for every(bj) ∈ p(1) and each family(Yij |aij ) of elementsin F . Suppose thatF satisfies(i) and(ii) in the definition of a filter and additionallyX|0 6∈ F as inclassical filter’s definition. Then it follows that there exists a minimal conditionaF > 0 such thatF|aF

is a conditional filter onX|aF .18

For every conditional filter baseB, inspection shows that

FB := Z : Y ⊑ Z ⊑ X for someY ∈ B

is a conditional filter, called the conditional filterconditionally generatedbyB.

Examples 3.14.1) The conditional trivial filter isX.

2) The stable collectionU(x) of all conditional neighborhoods of a conditional elementx of a conditionaltopological space is a conditional filter.

3) For any conditional subsetY of X on1, the collectionZ : Y ⊑ Z ⊑ X is a conditional filter. ♦

Proposition 3.15. LetX be a conditional set andB a stable collection of conditional subsets ofX on1.ThenB is a conditional filter base if, and only if, the corresponding stable collectionB of stable subsetsofX is a classical filter base onX .

17The conditional inclusion between the two stable collections of conditional subsets ofX is understood in the sense of theconditional inclusion of the generated conditional sets.

18 Indeed, suppose, for the sake of contradiction, that∧ai = 0. By de Morgan’s law, it holds∨aci = 1. By the well-orderingtheorem, choose(bi) ∈ p(1) such thatbi ≤ aci for eachi. Then

∑Yi|bi = X|0. SinceF is a stable collection of conditional

sets, it follows that∑

Yi|bi ∈ F which contradictsX|0 6∈ F .

15

Proof. By Lemma2.13, the equivalence is immediate from the definitions.

We next prove a conditional ultrafilter lemma.

Theorem 3.16. For every conditional filterF there exists a conditional ultrafilterU such thatF ⊑ U .

Proof. OrderF := G : G is a conditional filter withF ⊑ G by conditional inclusion. Let(Gi) be achain inF and setW := ⊔Gi. We show thatW is a conditional filter withF ⊑ W . LetZ ⊑ X be suchthatY ⊑ Z for someY ∈ W . ThenY =

∑

Yj |aj for some(aj) ∈ p(1) andYj ∈ Gij for eachj.For eachj, it holdsZ|aj ∈ Gij |aj becauseYj |aj ⊑ Z|aj andGij |aj is a conditional filter onX|aj . Bystability,Z|aj +X|acj ∈ W for all j, and thusZ =

∑

(Z|aj + X|acj)|aj ∈ W . Let Y,Z ∈ W whereY =

∑

Yj |aj andZ =∑

Zl|bl for some(aj), (bl) ∈ p(1), andYj ∈ Gij for eachj andZl ∈ Gil foreachl. Since(Gi) is a chain, it holdsYj ⊓ Zl is a conditional subset on1 in Gi for i = j ∨ l for everyj, l. By stability, it holds thatY ⊓ Z =

∑

(Yj ⊓ Zl)|aj ∧ bl is a conditional set on1 in W . HenceWis a conditional filter withF ⊑ W and therefore an upper bound for(Gi). The existence of a conditionalultrafilter follows by Zorn’s lemma.

Proposition 3.17. LetU be a conditional filter. Then the following are equivalent:

(i) U is a conditional ultrafilter.

(ii) If Y1 ⊔Y2 ∈ U for someY1,Y2 ⊑ X, thenY1|a+Y2|ac ∈ U where eithera = a1 or ac = a2,wherebyY1 is ona1 andY2 is ona2.

(iii) For everyY ⊑ X, it holdsY|a +Y⊏|ac ∈ U where eithera = a1 or ac = a2, wherebyY is ona1 andY⊏ is ona2.

(iv) For everyY ⊑ X such thatY ⊓U is on1 for everyU ∈ U , it holdsY ∈ U .

Proof. To show that(i) implies(ii ), letY1,Y2 ⊑ X be such thatY1 ⊔Y2 ∈ U . Since

Y1 ⊔Y2 = Y1|b1 + (Y1 ⊔Y2)|b2 +Y2|b3

whereb1 = a1 ∧ ac2, b2 = a1 ∧ a2 andb3 = a2 ∧ ac1, it holdsY1|b1,Y1|b2 ⊔Y2|b2,Y2|b3 ∈ U . Nowif Y1|b2 ∈ U , thena = b1 ∨ b2 = a1 yields the claim. OtherwiseF := Z ⊑ X : Z ⊔ Y1|b2 ∈ Uis a conditional filter such thatY2|b2 ∈ F . SinceU ⊑ F andU is a conditional ultrafilter, it holdsY2|b2 ∈ U . In that caseac = b2 ∨ b3 = a2 yields the assertion.

As for (ii ) implies(iii ), setY1 := Y andY2 := Y⊏. Then(iii ) follows sinceY ⊔Y⊏ = X ∈ U .

To show that(iii ) implies (i), let V be a conditional filter conditionally finer thanU andY ∈ V . Forthe sake of contradiction, supposeY 6∈ U . Then by assumptionY|a + Y⊏|ac ∈ U whereY⊏ ison ac > 0. HenceY⊏|ac ∈ U ⊑ V . However, sinceY|ac + X|a,Y⊏|ac + X|a ∈ V , it holds(Y|ac + X|a) ⊓ (Y⊏|ac + X|a) = X|a which contradicts the second property of a conditional filter.Thusac = 0, showing thatV = U .

Finally we show that(i) is equivalent to(iv). Assume(i) and letY ⊑ X be such thatY ⊓U is on1 foreveryU ∈ U . Inspection shows thatB := Y ⊓U : U ∈ U is a conditional filter base withU ⊑ FB.HenceU = FB, and thusY ∈ U . Conversely, letV be a conditional ultrafilter ofU and letY ∈ V . FromU ⊑ V it follows thatY ⊓U is on1 for everyU ∈ U . By assumption, one hasY ∈ U , and thereforeV = U .

16

Proposition 3.18. LetX andY be conditional sets,f : X → Y a conditional function,F a conditionalfilter onX andU a conditional ultrafilter onX. Thenf(F) := f(U) : U ∈ F is a conditional filterbase onY andf(U) a conditional ultrafilter base onY.

Proof. SinceF is a stable collection of conditional subsets ofX, it follows from the stability off thatf(F) is also a stable collection of conditional subsets ofX. It is immediate from the definitions, (2.4) and(2.6) that f(F) is a conditional filter base. Next suppose thatU is a conditional ultrafilter and denote byV the conditional filter conditionally generated byf(U). LetV ⊑ Y be ona1 andV⊏ on a2. By (2.5)and (2.7),

f−1(V) ⊔ f−1(V)⊏ = f−1(V ⊔V⊏) = f−1(Y) = X ∈ U .

SinceU is a conditional ultrafilter, Proposition3.17implies thatU := f−1(V)|b + f−1(V⊏)|bc ∈ Uwhere eitherb = b1 or bc = b2 wherebyf−1(V) is onb1 ≤ a1 andf−1(V⊏) is onb2 ≤ a2. By (2.7) and(2.8), it holds

f(U) = f(

f−1(V))

|b+ f(

f−1(V⊏))

|bc ⊑ V|b+V⊏|bc.

Sincef(U) ∈ f(U), one hasV|b + V⊏|bc ∈ V . Without loss of generality, assume thatb = b1 ≤ a1.By concatenating, we obtainV|a + V⊏|ac ∈ V wherea = a1. Proposition3.17 implies thatV is aconditional ultrafilter.

3.3 Conditional convergence

Definition 3.19. Let (X, T ) be a conditional topological space,F a conditional filter and(xi) a condi-tional net of conditional elements ofX. A conditional elementx of X is said to be a

(i) conditional limit pointof F if U(x) ⊑ F ;

(ii ) conditional cluster pointof F if x is a conditional element ofcl(Y) for everyY ∈ F and denoteLimF := ⊓cl(Y) : Y ∈ F;

(iii ) conditional limit pointof (xi) if for every conditional neighborhoodU of x there existsi0 suchthat eachxi is a conditional element ofU for everyi > i0;

(iv) conditional cluster pointof (xi) if for every conditional neighborhoodU of x and everyi thereexistsj > i such thatxj is a conditional element ofU.

We indicate byF → x thatx is the conditional limit point ofF . For a classical topologyT and a

classical filterF , denote byFT−→ x the convergence ofF to x and byLimF the set of all cluster

points ofF with respect toT .

Remark 3.20.As in the classical case, inspection shows that conditionalfilters and conditional nets are inone-to-one relation. The following Propositions3.21, 3.22and3.23are formulated in terms of conditionalfilters. Their respective analogues hold for conditional nets.

Proposition 3.21. Let (X, T ) be a conditional topological space andF a conditional filter. Then thefollowing are equivalent:

(i) The conditional elementx is a conditional element ofLimF .

(ii) There exists a conditional filterG conditionally finer thanF such thatG → x.

17

Proof. To show that(i) implies(ii) , letx be a limit point ofF . ThenV ⊓U : V ∈ U(x),U ∈ F is aconditional filter base of a conditional filterG conditionally finer thanF and for which holdsG → x. Toshow that(ii) implies(i), letG be a conditional filter conditionally finer thanF andG → x. ThenV ⊓Y

is on1 for all V ∈ U(x) andY ∈ F sinceV,Y ∈ G. Proposition3.7 implies thatx is a conditionalelement ofcl(Y) for all Y ∈ F which shows thatx is a conditional limit point ofF .

Proposition 3.22. Let X be a conditional set,G a conditional filter base,B a conditional topologicalbase, andG andB the corresponding stable collections of stable subsets ofX . Then

x ∈ LimG if, and only if, x is a conditional element ofLimG

GT

B

−−→ x if, and only if, G → x.

Proof. The assertions follow from the Propositions3.5, 3.7and3.15.

Proposition 3.23. Let (X, T ) and(X′, T ′) be conditional topological spaces andf : X → X′ a condi-tional function. Then the following are equivalent:

(i) The conditional functionf is conditionally continuous atx.

(ii) For every conditional filterF → x, it holdsf(F) → f(x).

Proof. By Proposition3.11, conditional continuity is equivalent to continuity. The claim follows fromthe respective classical result and Proposition3.22.

3.4 Conditional compactness

Let (X, T ) be a conditional topological space. Aconditional open coveringof X is a conditional family(Oi) of conditional open sets such thatX = ⊔Oi. A conditional family(Yi) of conditional subsets ofX has theconditional finite intersection propertyif19 ⊓16l6nYil is on 1 for every conditionally finitesubfamily(Yil)16l6n.

Definition 3.24. We say thatX is conditionally compactif for every conditional open covering(Oi)

there exists a conditionally finite subfamily(Oil)16l6n such thatX = ⊔16l6nOil .

Proposition 3.25. Let (X, T ) be a conditional topological space. Then the following are equivalent:

(i) X is conditionally compact.

(ii) Every conditional filter onX has a conditional cluster point.

(iii) Every conditional ultrafilter onX has a conditional limit point.

(iv) For every conditional family(Fi) of conditional closed subsets ofX with the conditional finiteintersection property,⊓Fi is on1.

Proof. The equivalence of(ii) and(iii) follows from Theorem3.16and Proposition3.21.

19Recall that⊔16l6n and⊓16l6n are understood as the conditional union and intersection over all conditional elementsl suchthat1 6 l 6 n.

18

To show that(i) implies (iv), let (Fi) be a conditional family of conditional closed subsets ofX withthe conditional finite intersection property. By contradiction, assume that⊓Fi is ona < 1. Let Oi :=

(Fi)⊏ ∈ T for eachi. Without loss of generality, assume thatOi is on1 for eachi. Otherwise, replace

Oi by Oi = Oi|ai + X|aci whereOi is on ai ≤ 1. By localization,X|ac is conditionally compactwith respect to the conditional topologyT |ac, and(Oi|ac) is a conditional open covering ofX|ac. Byassumption, there exists a conditionally finite subfamily(Oil |a

c)16l6n such thatX|ac = ⊔16l6nOil |ac.

It follows from de Morgan’s law

X|a = (X|ac)⊏ = (⊔16l6nOil |ac)⊏ = ⊓16l6n(Oil |a

c)⊏

= ⊓16l6n(Fil |ac +X|a) = (⊓16l6nFil) |a

c +X|a.

By the conditional finite intersection property,⊓16l6nFil is on1, and therefore(⊓16l6nFil)|ac is onac.

Thus it holdsX|a = (⊓16l6nFil)|ac +X|a if, and only if,ac = 0 which is the desired contradiction.

To show that(iv) implies(i), let (Oi) be a conditional open covering ofX. Let

b := ∨a : ⊔16l6nOil |a = X|a for some conditionally finite subfamily(Oil).

By consistency, the well-ordering theorem and stability,b is attained by some(Oil). By contradiction,suppose thatb < 1. Up to localization, we may assume thatbc = 1, otherwise the following argument isdone onbc < 1. Then for all conditionally finite subfamilies(Oil), it holds

(⊔16l6nOil) |a 6= X|a, for all 0 < a ≤ 1.

Hence(⊔16l6nOil)⊏= ⊓16l6nOil

⊏ = ⊓16l6nFil is on1, and thus(Fi) satisfies the conditional finiteintersection property. By assumption,⊓Fi is on 1. However, this implies that(⊓Fi)

⊏ = ⊔Oi 6= X,contrary to the assumption.

As for (ii) implies (iv), let (Fi) be a conditional family of conditional closed subsets ofX satisfyingthe conditional finite intersection property. ThenG := ⊓16l6nFil : (Fil)16l6n conditionally finite isa conditional filter base. By assumption, there exists a conditional elementx of Lim(F). Thus⊓Fi ison1.

To show that(iv) implies(ii) , letF be a conditional filter onX. Sincecl(⊓Yi) ⊑ ⊓cl(Yi), it followsthatcl(Y) : Y ∈ F is a stable collection of conditional closed subsets ofX fulfilling the conditionalfinite intersection property. HenceLimF is on1, and thusF has a conditional cluster point.

Proposition 3.26. Let (X, T ) and(X′, T ′) be conditional topological spaces,f : X → X′ a condition-ally continuous function, andY a conditional compact subset ofX on 1. Thenf(Y) is a conditionalcompact subset ofX′.

Proof. Let (Oi) be a conditional open covering off(Y), that isf(Y) ⊑ ⊔Oi. By (2.4), (2.5) and (2.8),one has

Y ⊑ f−1(f(Y)) ⊑ f−1(⊔Oi) = ⊔f−1(Oi).

By Proposition3.10, it follows that(f−1(Oi)) is a conditional open covering ofY. By assumption, thereexists a conditionally finite subfamily(f(Oil))16l6n such thatY ⊑ ⊔16l6nf(Oil). By (2.4), (2.5) and(2.8), one has

f(Y) ⊑ f(⊔16l6nf−1(Oil)) = ⊔16l6nf(f

−1(Oil)) ⊑ ⊔16l6nOil .

19

Proposition 3.27. Let (X, T ) be a conditionally compact space andY a conditionally closed subset ofX on1. ThenY is conditionally compact.

Proof. Without loss of generality, we may assume thatY⊏ is on 1, since otherwiseY|ac = X|ac isalready conditionally compact by localization, whereY⊏ is ona < 1. Let (Oi) be a conditional opencover ofY. Then(Oi) whereOi := Oi ⊔ Y⊏ is a conditional open cover ofX. By assumption,there exists a conditionally finite subfamily(Oil)16l6n conditionally coveringX. Thus(Oil)16l6n is aconditional open cover ofY.

We finish this section with a conditional Tychonoff’s theorem.

Theorem 3.28. Let (Xi, Ti) be a non-empty family of conditional topological spaces andletX =∏

Xi

be endowed with the conditional product topology. ThenX is conditionally compact if, and only if,Xi isconditionally compact for everyi.

Proof. Every conditional projectionpri : X → Xi is conditionally continuous. Therefore, ifX is condi-tionally compact, so isXi = pri(X) for everyi due to Proposition3.26. Conversely, assume thatXi isconditionally compact for eachi and letU be a conditional ultrafilter onX. It follows from Proposition3.18thatpri(U) is a conditional ultrafilter base onXi for eachi. SinceXi is conditionally compact,pri(U) → xi for some conditional elementxi of Xi for eachi due to Proposition3.25. Let Oi be aconditional open neighborhood ofxi for somei. Then there existsU ∈ U such thatpri(U) ⊑ Oi. By(2.4) and (2.8), one hasU ⊑ pr−1

i (Oi), and thuspr−1i (Oi) ∈ U . By stability ofU , it follows that any

conditional neighborhood ofx := (xi) is an element inU which shows thatU → x. Proposition3.25implies thatX is conditionally compact.

4 Conditional real numbers and metric spaces

Definition 4.1. We call a conditional setX together with a conditional function+ : X × X → X aconditional groupif (X,+) is a classical group. A conditional setX with two conditional functions+ : X ×X → X and· : X × X → X is aconditional ringif (X,+, ·) is a classical ring. We denoteby 0 and1 the conditional neutral elements for+ and ·, respectively.20 Let (X,+, ·) be a conditionalring and(X,6) a conditional totally ordered set. We say thatX is aconditional ordered ringif x < y

impliesx + z < y + z andx,y > 0 impliesx · y > 0 for all conditional elementsx,y, z of X. Aconditional ring(X,+, ·) is aconditional fieldif for every conditional elementx of X∗ := 0⊏ thereexists a conditional elementy of X∗ such thatx · y = y · x = 1.

For a conditional ordered field(K,+, ·,6), defineK+ := x : x > 0 andK++ := x : x > 0. Theconditional absolute value|·| : K → K+ is generated by the stable function|x| := maxx,−x.

Examples 4.2.1) Let(S,+, ·) be a classical ring. The conditional functions generated by+ and· definea conditional ring structure on the conditional setS generated byS in the sense of Example2.35). Forinstance, the conditional ring of conditional rational numbers(Q,+, ·) is generated by the ring of rationalnumbers(Q,+, ·). Inspection shows thatQ is a conditional ordered field where the conditional order isthe one defined in Examples2.161).

2) LetQN be the conditional set of conditional sequences of conditional elements ofQ. OnQN define

(qn) + (pn) := (qn + pn), (qn) · (pn) := (qn · pn),

(qn) 6 (pn) wheneverqn 6 pn for eachn.

20Whenever there is no risk of confusion, we use+ for addition and concatenations, and0, 1 denote the neutral elements and thedistinguished elements0, 1 ∈ A.

20

Inspection shows that(QN,+, ·,6) is a conditional ordered ring. ♦

EndowQ with the conditional Euclidean topology, see Example3.2. A conditional sequence(qn) ofconditional elements ofQ is said to beconditionally Cauchyif for every conditional elementr of Q++

there existsn0 such that|qn − qm| < r for all m,n > n0. Denote byC the conditional subset ofQN consisting of all conditional Cauchy sequences. Let(pn) ∼ (qn) whenever(qn − pn) → 0 be aconditional equivalence relation onC. OnR := C/ ∼ define

[(qn)] + [(pn)] := [(qn) + (pn)], [(qn)] · [(pn)] := [(qn) · (pn)],

[(pn)] 6 [(qn)] whenever for allr of Q++ there existsn0 such thatqn − pn > −r for all n > n0.

Inspection shows that(R,+, ·,6) is a conditional ordered field.

Definition 4.3. We callR = (R,+, ·,6) the conditional ordered field ofconditional real numbers.

It can be checked that

(i) R is conditionally Dedekind complete;

(ii) for every conditional elementx of R there exists a conditional elementn of N such thatn > x;

(iii) for conditional elementsx, r of R such thatr > 0, let Br(x) = y : |x− y| < r. The condi-tional collectionB of conditional setsBr(x) : x, r of R with r > 0 is a conditional topologicalbase of a conditional Hausdorff topology.R endowed withT B is conditionally separable andcomplete. We callT B theconditional Euclidean topologyonR.

The following theorem connects the conditional analysis inL0-modules [8, 10, 13] to conditional settheory. For a proof of the isomorphism ofL0 and the real numbers in the universe of Boolean-valued setsover the measure algebra associated to aσ-finite measure space see [2, Theorem 7.1]. The conditional setL0 is constructed in Example2.34), and its conditional topology is defined in Example3.6.

Theorem 4.4. Let A be the measure algebra associated to aσ-finite measure space(Ω,F , µ). Thenthere exists a conditional bijection fromR toL0.

Proof. For∑

qn|an ∈ Q, one has∑

n≥1 qn1An:= limn→∞(q11A1

+ . . . + qn1An) ∈ L0 wherean =

[An] for eachn. Letx be a conditional element ofL0. SinceL0 is conditionally separable, there exists aconditional sequence(qn) of conditional elements ofL0 with qn =

∑

m≥1 qn,m1An,mwhere(qn,m) is

a sequence inQ such thatqn → x, see [21, Lemma 5.3.2]. Conversely, if(qn) is a conditional Cauchysequence of conditionally rational-valued functions, then (qn) is a Cauchy net inL0 endowed with theL0-topology. By a conditional Bolzano-Weierstraß theorem, see [15, Lemma 1.64] or [8, Theorem 3.8],one has that(qn) converges in theL0-topology.21 From Proposition3.22it follows that(qn) conditionallyconverges tox. Thenf : R → L0 defined by the stable functionf(qn) = lim(

∑

m≥1 qn,m1An,m) is a

conditional function. It is conditionally injective sinceL0 is conditionally Hausdorff, and conditionallysurjective due to the conditional separability ofL0.

Definition 4.5. Let X be a conditional set. Aconditional metricis a conditional functiond : X×X →R+ such that

(i) d(x,y) = 0 if, and only if,x = y,

(ii) d(x,y) = d(y,x) for all conditional elementsx, y of X,

21In the cited results the convergence is almost everywhere which implies convergence inL0-topology for stable sequences.

21

(iii) d(x, z) 6 d(x,y) + d(y, z) for all conditional elementsx,y, z of X.

The pair(X,d) is called aconditional metric space.

Given a conditional metric space(X,d), we define a conditional topological baseB consisting of con-ditional ballsBr(x) and conditional Cauchy sequences similarly to the respective definitions forQ, seeExample3.2. Inspection shows that the conditional topologyT B is conditionally Hausdorff and firstcountable. We say that(X,d) is conditionally complete, if every conditional Cauchy sequence has aconditional limit, andconditionally sequentially compact, if every conditional sequence has a conditionalcluster point. A conditional subsetY of X on1 is conditionally totally bounded, if for every conditionalelementr of R++ there exists a conditionally finite family(xl)16l6n of conditional elements ofY,called aconditionalr-net, such that⊔16l6nBr(xl) = Y. A conditionally totally bounded metric spaceis conditionally separable. Indeed, for every conditionalelementr of Q++ choose(xr

l )16l6mrsuch that

⊔16l6mrBr(x

rl ) = X. Then⊔r of Q++

xrl : 0 6 l 6 mr is a conditionally countable dense subset by

Proposition2.25.We characterize conditional compactness in conditional metric spaces by a conditional Borel-Lebesgue

theorem.

Theorem 4.6. For a conditional metric space(X,d), the following are equivalent:

(i) X is conditionally compact.

(ii) If (Fn) is a conditionally decreasing22 family of conditionally closed sets inX, then⊓Fn is on1.

(iii) X is conditionally sequentially compact.

(iv) X is conditionally complete and totally bounded.

Proof. As for (i) implies (ii) , note that⊓16l6mFnl= Fnm

for any conditionally finite subfamily(Fnl

)16l6m. Thus⊓16l6mFnlis on1. Proposition3.25implies that⊓Fn is on1.

To show that(ii) implies(iii) , let (xn) be a conditional sequence of conditional elements ofX. For eachn ∈ N , setEn := xm : m ≥ n. Then(En) is a stable family of stable subsets ofX , and thus(En)

is a conditional family of conditional subsets ofX. Let Fn := cl(En) for each conditional elementnof N. Then(Fn) is a conditionally decreasing family of conditional closedsets. By assumption, thereexists a conditional elementx of ⊓Fn. Now setxn1

= x1. Supposing that we have already chosenxn1

, . . . ,xnk−1, choose a conditional elementxnk

of Fnk−1+1 such thatd(xnk,x) < 1/k. For each

k =∑

ni|ai ∈ N , setxnk:=

∑

xni|ai. Inspection shows that(xnk

) is a conditional subsequence of(xn) conditionally converging tox.

We show that(iii) implies(iv). Conditional completeness follows by the conditional triangle inequality.As for the conditional total boundedness, suppose for the sake of contradiction thatb := ∨M > 0 where

M = a : ⊔16l6nBr(xl)|a′ 6= X|a′ for all 0 < a′ ≤ a for somer > 0 and all(xl)16l6n of X.

Note that ifa ∈ M , thena′ ∈ M whenevera′ ≤ a. IndexM by (ai). By the well-ordering theorem,there exists(bi) ∈ p(b) with bi ≤ ai for all i. Thenbi ∈ M for eachi. Let ri > 0 satisfy the condition inM for bi for eachi. By stability,b is attained inM for r =

∑

ri|bi. Without loss of generality, assume

22That is,m 6 n impliesFn ⊑ Fm.

22

thatb = 1. Letx1 be any conditional element ofX. Suppose we have found a conditionally finite family(xl)16l6n such thatd(xl1 ,xl2) > r for all 1 6 l1, l2 6 n with l1 ⊓ l2 = N|0. Let

c := ∨a : d(xl,x)|a > r|a for somex of X and for all1 6 l 6 n.

By consistency, the well-ordering theorem and stability,c is attained by somexn+1 := x. It holdsc = 1,since otherwise there existsa > 0 such that⊔16l6nBr(xl)|a = X|a which contradicts the maximalityof b. For1 6 l 6 n+ 1, putxl :=

∑

xni|ai wherel =

∑

ni|ai. By induction, we obtain a conditionalsequence(xn) such thatd(xn1

,xn2) > r whenevern1 ⊓ n2 = N|0. By construction,(xn) does not

have a conditionally converging subsequence. Indeed, if there exists a conditional subsequence(xnk)

conditionally converging to somex of X, then there existsk0 of N such thatd(xnk,x) < r/2 for all

k > k0. By the conditional triangle inequality, one hasd(xnk0+1,xnk0+2

) < r contrary to the definitionof (xn).

To show that(iv) implies(i), suppose, by contradiction, thatb := ∨M > 0 where

M = a : ⊔16l6nUil |a′ 6= X|a′ for all 0 < a′ ≤ a and(Uil ) of some conditional open covering(Ui).

By consistency, the well-ordering theorem and stability,b is attained by some(Ui). By localization, wemay assume thatb = 1. Now for r = 1/2 there exists a conditional1/2-net(x1

l )16l6n1by assumption.

Let

c1 := ∨a : ⊔16k6mUik |a′ 6= B1/2(x

1l )|a

′ for all 0 < a′ ≤ a and(Uik)16k6m of (Ui) for somex1l .

By consistency, stability and the well-ordering theorem,c1 is attained by somey1 := x1l . By as-

sumption, it must holdc1 = 1. Indeed, ifcc1 > 0, then for allx1l there exists(Uik)16k6m such that

⊔16k6mUik |cc1 = B1/2(x

1l )|c

c1. Proposition2.25implies thatX|cc1 is conditionally covered by a condi-

tionally finite subfamily of(Ui|cc1) which contradicts the maximality ofb.

Next for r = 1/4, let (x2l )16l6n2

be a conditional1/4-net, and setc = ∨M whereM is the collectionof all a ∈ A such that there existsx2

l with B1/2(y1) ⊓B1/4(x2l ) is on1 and

⊔16k6mUik |a′ 6= B1/4(x

2l )|a

′ for all 0 < a′ ≤ a and(Uik )16k6m of (Ui).

By consistency, the well-ordering theorem and stability,c2 is attained by somey2 := x2l . Moreover, one

hasc2 = 1. Indeed, since(B1/4(x2l ))16l6n2

is a conditional family, for each conditional elementx ofB1/2(y1) there existsx2

l such thatx is a conditional element ofB1/4(x2l ) due to Lemma2.13. Thus if

cc2 > 0, then for allx2l such thatB1/2(y1)⊓B1/4(x

2l ) is on1 there exists a conditionally finite subfamily

(Uik) of (Ui) such that⊔Uik |cc = B1/4(x

2l )|c

c2. By Proposition2.25, this contradicts the maximality

of b.

We continue analogously: at then-th stage, withr = 1/2n let yn be any conditional element of thatconditional1/2n-net such thatB1/2n−1(yn−1) ⊓ B1/2n(yn) is on1, and having the property that if aconditional finite subfamily of(Ui|a) conditionally coversB1/2n(yn)|a, thena = 0.

For n =∑

ni|ai, setyn =∑

yni|ai. By construction,(yn) is a conditional Cauchy sequence of

conditional elements ofX. By assumption,(yn) conditionally converges to some conditional elementy

of X. Since(Ui) conditionally coversX, there exists somei0 such thaty is a conditional element ofUi0 by Lemma2.13. SinceUi0 is conditionally open there existsr > 0 such thatBr(y) ⊑ Ui0 . Bythe definition ofy, there existsn of N such thatd(yn,y) < r/2 and1/2n < r/2. It follows from theconditional triangle inequality thatB1/2n(yn) ⊑ Br(y) ⊑ Ui0 . But this contradicts the fact that theredoes not exist a conditionally finite subfamily of(Ui) which is a conditional covering ofB1/2n(yn).

23

5 Conditional topological vector spaces

Definition 5.1. A conditional groupX is aconditional vector spaceif there exists a conditional function· : R ×X → X such thatX is anR-module in the classical sense. A conditional subsetY of X on 1

is aconditional subspaceof X if Y + Y ⊑ Y andrY ⊑ Y for every conditional elementr of R. Aconditional functionf : X → R is conditionally linearif f(rx+ y) = rf(x) + f(y) for all conditionalelementsx,y of X and every conditional elementr of R.

Definition 5.2. Let X be a conditional vector space. LetY be a conditional subset ofX on1. Define

conv(Y) :=

∑

16l6n

rlyl : (yl)16l6n of Y, 0 6 rl 6 1,∑

16l6n

rl = 1, n of N

.23

We say thatY is

• conditionally convexif Y = conv(Y),

• conditionally absorbingif for any conditional elementx of X there existsrx > 0 such thatrx is aconditional element ofY for everyr of R with |r| 6 rx,

• conditionally circledif ry is a conditional element ofY for every conditional elementy of Y andall conditional elementsr of R with |r| 6 1.

A conditional functionf : X → R is conditionally convexif

f(rx+ (1− r)y) 6 rf(x) + (1− r)f(y)

for all conditional elementsx,y of X and0 6 r 6 1.

We prove a conditional Hahn-Banach theorem.

Theorem 5.3. Let X be a conditional vector space,Y a conditional subspace ofX and f : Y → R

a conditional linear function. Iff(x) 6 g(x) for all conditional elementsx of Y for some conditionalconvex functiong : X → R, then there exists a conditional linear functionf : X → R such thatf(x) 6 g(x) for all conditional elementsx of X and f(x) = f(x) for all conditional elementsx ofY.

Proof. Let E be the collection of all pairs(h,H) whereH is a conditional subspace ofX with Y ⊑ H

andh : H → R is a conditionally linear function such thath = f on Y andh 6 g on H. Definea partial order onE by (h,H) ≤ (h′,H′) wheneverH ⊑ H′ andh′ = h on H. Let (hi,Hi) be achain inE . By Proposition2.4, it holds(h,H) ∈ E whereH := ⊔Hi andh : H → R is defined byh(x) =

∑

hij (xj)|aij for everyx =∑

xj |aj of H where(aj) ∈ p(1) andxj of Hij for eachj. ByZorn’s lemma, there exists a maximal element(f , H) in E .

By contradiction, suppose thatH⊏ is ona > 0. By localization, we may assume thata = 1. Pick somev of H⊏ and putH = x+ rv : x of H, r of R. Inspection shows thatH is a conditional subspace ofX with Y ⊑ H ⊏ H ⊑ X. Every conditional elementy of H is of the formy = x+ rv for uniquexof H andr of R. Indeed, lety = x+ rv = x+ rv and suppose thatb = ∨a : r|a = r|a < 1. Since(r− r)v = x− x is a conditional element ofH, it holds thatv ⊓ H is onbc > 0 which contradicts thechoice ofv. Henceb = 1, and thusr = r andx = x.

23We understand∑

16l6nxl :=

∑(∑ni

l=1xl)|ai wheren =

∑ni|ai.

24

Any linear extensionf of f to H has to fulfill f(x+ rv) = f(x) + rf(v) for all x of H andr of R. Itis enough to findw of R such thatf(x) + rw 6 g(x+ rv) for all x of H andr of R. In this case, thereis (a1, a2, a3) ∈ p(1) such thatr|a1 > 0|a1, r|a2 < 0|a2 andr|a3 = 0|a3. Thus

w|a1 6

(

1

r[g(x+ rv) − f(x)]

)

|a1, (5.1)

w|a2 6

(

−1

r[f (x)− g(x+ rv)]

)

|a2, (5.2)

f(x)|a3 6 g(x)|a3, (5.3)

for everyx of H. The relation (5.3) is fulfilled by the definition off . By inspection, the relations (5.1)and (5.2) hold for a conditional elementr of R if, and only if,

1

p[f(x)− g(x − pv)] 6 w 6

1

q[ g(y + qv) − f (y)] (5.4)

for all conditional elementsx,y of H andp,q of R++. Thus (5.4) is fulfilled for some conditionalelementr of R which is the desired contradiction.

Definition 5.4. A conditional vector spaceX endowed with a conditional topologyT is a conditionaltopological vector spaceif the conditional functions+ : X × X → X and · : R × X → X areconditionally continuous. We call a conditional subsetY of X on1 conditionallyT -boundedif for everyconditional neighborhoodU of 0 there existsr > 0 such thatY ⊑ rU. For a conditional topologicalvector spaceX, denote byX′ the conditional vector space of all conditionally continuous and linearfunctionsf : X → R.

A conditional topological vector spaceX is conditionally locally convexif X has a conditional neigh-borhood base of0 consisting of conditionally convex sets.

Inspection shows that a conditional topological vector space has a conditional neighborhood base of0

consisting of conditionally closed, absorbing and circledconditional sets.Separation theorems for locally convex topologicalL0-modules are proved in [13, Theorem 2.6 and

2.8]. The following theorem is the respective analogue for conditional locally convex topological vectorspaces. Its proof technique is similar to [13].

Theorem 5.5. Let X be a conditional locally convex topological vector space and C1,C2 two condi-tionally convex subsets ofX on1 such thatC1 ⊓C2 = X|0.

(i) If C1 is conditionally open, then there exists a conditionally continuous linear functionf : X → R

such thatf(x) < f(y) for every conditional elementx of C1 andy ofC2.

(ii) If C1 is conditionally compact andC2 conditionally closed, then there exists a conditionally con-tinuous linear functionf : X → R and a conditional elementr ofR++ such that

f(x) + r < f(y) for every conditional elementx ofC1 andy ofC2.

Definition 5.6. Two conditional vector spacesX andY form aconditional dual pair, denoted by〈X,Y〉,if there exists a conditional function〈·, ·〉 : X×Y → R such that

25

(i) if x 7→ 〈x,y〉 for every fixed conditional elementy of Y andy 7→ 〈x,y〉 for every fixed condi-tional elementx of X are conditionally linear;

(ii) if 〈·,y〉 = 0 and〈x, ·〉 = 0 imply y = 0 andx = 0, respectively.

For a conditional dual pair〈X,Y〉, we denote byσ(X,Y) andσ(Y,X) the conditional initial topologiesfor (〈·,y〉)y of Y and (〈x, ·〉)x of X, respectively. It can be checked that(X, σ(X,Y))′ = X′. For aconditional locally convex topological vector spaceX, it follows from Theorem5.3 that 〈X,X′〉 is aconditional dual pair with respect to the conditional function (x,x′) 7→ x′(x).

Definition 5.7. Let 〈X,Y〉 be a conditional dual pair andZ a conditional subset ofX on 1. The condi-tional subsetZ := y : 〈x,y〉 6 1 for all x of Z24 of Y is called theconditional polarof Z.

Proposition 5.8. Let 〈X,Y〉 be a conditional dual pair,Z andW two conditional subsets ofX on 1,and(Zi) a non-empty family of conditional subsets ofX each of which is on1. Then it holds:

(i) If Z ⊑ W, thenW ⊑ Z.

(ii) If r is a conditional element ofR∗ = 0⊏, then(rZ) = (1/r)Z.

(iii) (⊔Zi) = ⊓Z

i .

(iv) Z is conditionally convex,σ(Y,X)-closed and0 is a conditional element of it.

Proof. The assertions(i), (ii) and(iii) are immediate from the definitions. Proposition3.23implies asser-tion (iv).

The previous proposition together with Theorem5.5yield a conditional Bipolar theorem.

Theorem 5.9. Let 〈X,Y〉 be a conditional dual pair andZ a conditional subset ofX on1. ThenZ =

cl(conv(Z ⊔ 0)). Thus ifZ is conditionally convex,σ(X,Y)-closed and0 is a conditional element ofit, thenZ = Z.

Next we prove a conditional Banach-Alaoglu theorem.

Theorem 5.10. Let X be a conditional locally convex topological vector space and U a conditionalneighborhood of0. ThenU is conditionallyσ(X′,X)-compact.

Proof. The conditionalσ(X′,X)-topology is the conditional topology of conditional pointwise conver-gence inX and a conditional relative topology of the conditional product topologyT onRX.25 Withoutloss of generality, we may assume thatU is conditionally convex and circled due to Proposition5.8. Theconditional function

Φ : (X′, σ(X′,X)) → (RX, T ), Φ(x′)(x) = 〈x′,x〉

is conditionally injective and its conditional inverse function

Φ−1 : (RX, T ) → (X′, σ(X′,X))

conditionally continuous. By Proposition3.26, it is enough to show thatΦ(U) is conditionally compact.To this end, letx be a conditional element ofX. SinceU is conditionally absorbing, there existsrx > 0

24Note thatZ is on1 since0 is a conditional element of it.25As in the classical case, one identifiesX′ with a conditional subspace ofXR.

26

such thatx is a conditional element ofrxU. Forx being a conditional element ofU, chooserx 6 1.SinceU is conditionally circled, it holds|〈x′,x〉| 6 rx for all conditional elementsx′ of U. PutKx =

r : |r| 6 rx. It follows from Theorem4.6thatKx is conditionally compact. By Theorem3.28,

K := f of RX : f(x) of Kx for all x of X

is conditionally compact. By Proposition3.27, it remains to show thatΦ(U) is conditionally closed.Let f be a conditional element ofcl(Φ(U)). Since the conditional pointwise limit of a conditional netof conditional linear functions is conditionally linear (due to the conditional continuity of conditionaladdition and scalar multiplication),f is conditionally linear. Sincerx 6 1 for x of U, it follows thatf(U) ⊑ r : |r| 6 1. Therefore,f is a conditional element ofΦ(X′). The remaining assertion followsfrom the fact thatU is conditionallyσ(X′,X)-closed.

Definition 5.11. Let X be a conditional vector space. Aconditional normis a conditional function‖·‖ : X → R+ such that

(i) ‖x‖ = 0 if, and only if,x = 0,

(ii) ‖rx‖ = |r| ‖x‖ for all conditional elementsx of X andr of R,

(iii) ‖x+ y‖ 6 ‖x‖ + ‖y‖ for all conditional elementsx,y of X.

A conditional vector space together with a conditional normis called aconditional normed vector space.

For a conditional norm‖·‖ : X → R, the conditional functiond : X ×X → R defined byd(x,y) =‖x− y‖ is a conditional metric. A conditional normed vector space(X, ‖.‖) is called aconditionalBanach spaceif (X,d) is conditionally complete. For a conditionally linear operatorT : X → R, theconditional operator norm‖T‖′ is defined by‖T‖′ := sup |T(x)| : x of X, ‖x‖ = 1. By inspection,if X is conditionally Banach, then(X′, ‖·‖′) is so.

We close this paper by a conditional Krein-Šmulian theorem.A conditional version of this theorem formodules overL∞ is proved in Eisele and Taieb [12, Theorem 11.1]. Forr > 0, let

C′r = x′ : ‖x′‖

′6 r, C′

r(y′) = x′ : ‖x′ − y′‖

′6 r for eachy′ of X′.

In particular, we denote byC′ := C′1 andC := x : ‖x‖ 6 1.

Theorem 5.12. LetX be a conditional Banach space andY a conditionally convex subset ofX′ on 1.ThenY is conditionallyσ(X′,X)-closed if, and only if,Y⊓C′

n is conditionallyσ(X′,X)-closed for allconditional elementsn ofN.

Proof. If Y is conditionallyσ(X′,X)-closed, then its conditional intersection with eachC′n is so, since

C′n is conditionallyσ(X′,X)-closed.As for the converse, note first thatY is conditionally norm-closed. Indeed, for a conditional sequence

(x′n) of conditional elements ofY such thatx′

n → x′, it follows from the conditional triangle inequalitythat there is somem of N such that(x′

n) is a conditional sequence of conditional elements ofY ⊓C′

m. Since conditional norm-convergence implies conditionalσ(X′,X)-convergence26 and sincex′n

is a conditional element ofY ⊓ C′m which is conditionallyσ(X′,X)-closed by assumption,x′ is a

conditional element ofY. FurtherY ⊓ C′r(y

′) is conditionallyσ(X′,X)-closed for all conditionalelementsr of R++ andy′ of X′. Indeed, there existsn of N such thatC′

r(y′) ⊑ nC′. By assumption,

Y ⊓C′r(y

′) = (Y ⊓ nC′) ⊓C′r(y

′) is conditionallyσ(X′,X)-closed.

26This follows from|〈xn,x′〉 − 〈x,x′〉| 6 ‖x′‖′ ‖xn − x‖.

27

Up to conditional addition ofY by a conditional element, which is a conditionally continuous oper-ation, we may assume that0 is a conditional element ofY. For each conditional elementn of N, putYn = Y ⊓ 2nC′. ThenYn is conditionally convex,σ(X′,X)-closed and0 is a conditional elementof Yn. It follows from Theorem5.9 thatYn = Y

n . PutZ = ⊓Yn. Note thatZ is on1 since0 is a

conditional element of eachYn. We will show thatY = Z which will imply thatY is conditionally

σ(X′,X)-closed by Proposition5.8. SinceZ ⊑ Yn, it follows thatYn ⊑ Z from Proposition5.8which

in turn impliesY = ⊔Yn ⊑ Z. The following steps will establishZ ⊑ Y.

Step 1 : We show thatYn ⊑ Y

n+1+2−nC for everyn of N. To this end, letx be a conditional elementof (Y

n+1 + 2−nC)⊏. By Theorem5.5, it holdsx′(x) > 1 > supx′(y) : y of Yn+1 + 2−nC

for somex′ of X′. Since2−(n+1)C ⊑ Yn+1 andx′(3 · 2−(n+1)C) = x′(2−(n+1)C+ 2−nC) ⊑

x′(Yn+1 + 2−nC) by (2.4), it holds‖x′‖′ 6 2

32n. Since

supy : y of Yn+1 + 2−nC = supy : y of Y

n+1+ supy : y of 2−nC,

one hassupy : y of Yn+1 6 1− 2−n ‖x′‖′. Chooser to be the conditional minimum of1/3 and

2−n ‖x′‖′. Puty′ := 11−r

x′. Fromr < 2−n ‖x′‖′ andsupy : y of Yn+1 6 1− 2−n ‖x′‖′, it follows

thatsupy : y of Yn+1 6 1, that is,y′ is a conditional element ofY

n+1 = Yn+1. By the choice ofr,we get‖y′‖ 6 2n, and thereforey′ is a conditional element ofYn. Sincey′(x) > 1, it follows thatx isa conditional element of(Yn

)⊏.

Step 2 : It is shown thatYn ⊑ Z + 2−(n−1)C for every conditional elementn of N. For eachn of

N, we pick inductively, using the result ofStep 1, xn+1,xn+2, . . . such thatxm is a conditional elementof Y

m and‖xm − xm+1‖ 6 2−m for m = n,n+ 1, . . .. For a conditional elementm of N withm =

∑

mi|ai where(mi) is a family inn, n + 1, . . . and(ai) ∈ p(1), putxm :=∑

xmi|ai. Since

(Yn) is a conditional family, eachxm is a conditional element ofY

m. By construction,(xm) is aconditional Cauchy sequence. By the conditional completeness ofX, there exists a conditional elementx of X such thatxm → x. EachY

n is conditionally norm-closed. Thusx is a conditional element ofZsince(xm) is a conditional element of eachY

m. Moreover, it holds‖x− xn‖ 6∑∞

j=n12j 6

12n−1 .

Step 3 : We show thatY = ⊓r>0(1+ r)Y. SinceY is conditionally convex and0 is a conditionalelement ofY, the conditional inclusion⊑ is immediate. Conversely, forx′ of ⊓r>0(1+ r)Y, settingx′n = n/(1+ n)x′ for n of N, defines a conditional sequence of conditional elements ofY such that

‖x′ − x′n‖

′6 ‖x′‖′ /(n+ 1) → 0. Hencexn → x, and thusx is a conditional element ofY sinceY is

conditionally norm-closed.

Step 4 : We show thatZ ⊑ ⊓r>0(1+ r)Y. It holdsx+ y = (1+ r)( x1+r

+ (1− 11+r

)yr) for all x,y

of X and somer > 0. FromStep 2it follows thatYn ⊑ (1+ r)conv(Z ⊔ 2n−1/rC). It follows from

Proposition5.8that1/(1+ r)(Z ⊓ 2n−1rC′) ⊑ Yn ⊑ Y for everyn of N. By taking the conditionalunion over alln of N, we obtain1/(1+ r)Z ⊑ Y for everyr > 0, and thusZ ⊑ ⊓r>0(1+ r)Y = Y

by means ofStep 3.

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