Inclusion hyperspaces and capacities on Tychonoﬀ spaces: functors and monads

8/6/2019 Inclusion hyperspaces and capacities on Tychonoff spaces: functors and monads

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arXiv:1008

.2926v1

[math.CT]17Aug2010

Inclusion hyperspaces and capacities on Tychonoff

spaces: functors and monads6

Oleh Nykyforchyna,, Dusan Repovsb

aDepartment of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian

National University, Shevchenka 57, Ivano-Frankivsk, Ukraine, 76025bFaculty of Mathematics and Physics, and Faculty of Education, University of Ljubljana,

P.O. Box 2964, Ljubljana, Slovenia, 1001

Abstract

The inclusion hyperspace functor, the capacity functor and monads for thesefunctors have been extended from the category of compact Hausdorff spaces tothe category of Tychonoff spaces. Properties of spaces and maps of inclusionhyperspaces and capacities (non-additive measures) on Tychonoff spaces areinvestigated.

Keywords: Inclusion hyperspace, capacity, functor, monad (triple), Tychonoffspace2000 MSC: 18B30

Introduction

The category of compact Hausdorff topological spaces is probably the most

convenient topological category for a categorical topologist. A situation is usualwhen some results are first obtained for compacta and then extended with mucheffort to a wider class of spaces and maps, see e.g. factorization theoremsfor inverse limits [13]. Many classical construction on topological spaces leadto covariant functors in the category of compacta, and categorical methodsproved to be efficient tools to study hyperspaces, spaces of measures, symmetricproducts etc [17]. We can mention the hyperspace functor exp [15], the inclusionhyperspace functor G [8], the probability measure functor P [5], and the capacityfunctor M which was recently introduced by Zarichnyi and Nykyforchyn [18] tostudy non-additive regular measures on compacta.

Functors exp, P, G, M have rather good properties. The functors exp and Pbelong to a defined by Scepin class of normal functors, while G and M satisfy all

6This research was supported by the Slovenian Research Agency grants P1-0292-0101, J1-9643-0101 and BI-UA/09-10-002, and by the Ministry of Science and Education of Ukraineproject M/113-2009.

Corresponding authorEmail addresses: [email protected] (Oleh Nykyforchyn),

[email protected] (Dusan Repovs)

Preprint submitted to Elsevier December 31, 2010
http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1http://lanl.arxiv.org/abs/1008.2926v1


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the map exp f : exp X exp Y, which sends each non-empty closed subset Fof X to its image f(Y), is continuous. Thus we obtain the hyperspace functor

exp : Comp C omp.A non-empty closed with respect to the Vietoris topology subset F exp X

is called an inclusion hyperspace if A B exp X, A F imply B F. Theset GX of all inclusion hyperspaces on the space X is closed in exp2 X, henceis a compactum with the induced topology if X is a compactum. This topologycan also be determined by a subbase which consists of all sets of the form

U+ = {F GX | there is F F, F U},

U = {F GX | F U = for all F F },

with U open in X. If the map Gf : GX GY for a continuous map f : X Yof compacta is defined as Gf(G) = {B

clY | B f(A) for some A F },

F GX, then G is the inclusion hyperspace functor in Comp.We follow a terminology of [18] and call a function c : exp X {} I acapacity on a compactum X if the three following properties hold for all closedsubsets F, G of X :

(1) c() = 0, c(X) = 1;

(2) if F G, then c(F) c(G) (monotonicity);

(3) ifc(F) < a, then there exists an open set U F such that for any G Uwe have c(G) < a (upper semicontinuity).

The set of all capacities on a compactum X is denoted by M X. It was shownin [18] that a compact Hausdorff topology is determined on M X with a subbasewhich consists of all sets of the form

O(F, a) = {c M X | c(F) < a},

where F cl

X, a R, and

O+(U, a) = {c M X | c(U) > a} =

{c M X | there exists a compactum F U, c(F) > a},

where U op

X, a R. The same topology can be defined as weak topology,

i.e. the weakest topology on M X such that for each continuous function :X [0;+) the correspondence which sends each c MX to the Choquetintegral [3] of w.r.t. c

X

(x) dc(x) = +

0

c({x X | (x) a}) da

is continuous. If f : X Y is a continuous map of compacta, then the mapMf : MX MY is defined as follows : M f(c)(F) = c(f1(F)), for c M Xand F

clY. This map is continuous, and we obtain the capacity functor M in

the category of compacta.

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A monad F in a category C is a triple (F, F, F), with F : C C a functor,F : 1C F and F : F

2 F natural transformations, such that FX

FF X = FX F FX = 1FX , FX F FX = FX FF X for all objectsX of C. Then F, F, F are called resp. the functorial part, the unit and themultiplication of F. For the inclusion hyperspace monad G = (G, G, G) thecomponents of the unit and the multiplication are defined by the formulae [11]:

GX(x) = {F exp X | F x}, x X,

and

GX(F) = {F exp X | F

H for some H F}, F G2X.

In the capacity monad M = (M, M, M) [18] the components of the unitand the multiplication are defined as follows:

M(x)(F) =

1, x F,

0, x / F,x X, F

clX,

and

MX(C)(F) = sup{ I | C ({c MX | c(F) }) }, C M2, F

clX.

An internal relation between the inclusion hyperspace monad and the capacitymonad is presented in [18, 9].

It is well known that the correspondence which sends each Tychonoff spaceX to its Stone-Cech compactification X is naturally extended to a functor : Tych Comp. For a continuous map f : X Y of Tychonoff spaces

the map f : X Y is the unique continuous extension of f. In factthis functor is left adjoint [7] to the inclusion functor U which embeds Compinto Tych. The collection i = (iX)XObTych of natural embeddings of allTychonoff spaces into their Stone-Cech compactifications is a unique naturaltransformation 1Tych U (a unit of the adjunction, cf. [7]).

In this paper monotonic always means isotone.

2. Inclusion hyperspace functor and monad in the category of Ty-chonoff spaces

First we modify the Vietoris topology on the set exp X for a Tychonoff spaceX. Distinct closed sets in X have distinct closures in X, but the map eexpX

which sends each F exp X to ClX F exp X generally is not an embed-ding when the Vietoris topology are considered on the both spaces, although iscontinuous. It is easy to prove :

Lemma 2.1. LetX be a Tychonoff space. Then the unique topology on exp X,such that eexpX is an embedding into exp X with the Vietoris topology, is

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determined by a base which consists of all sets of the form

U1, . . . , U k = {F exp X | F is completely separated fromX\ (U1 Uk), F Ui = , i = 1, . . . , k},

with all Ui open in X.

Observe that our use of the notation . . . differs from its traditional mean-ing [15], but agrees with it if X is a compactum. Hence this topology coincideswith the Vietoris topology for each compact Hausdorff space X, but may beweaker for noncompact spaces. The topology is not changed when we take aless base which consists only of U1, . . . , U k for Ui

opX such that U2 Uk

is completely separated from X \ U1. We can also equivalently determine ourtopology with a subbase which consists of the sets

U = {F exp X | F is completely separated from X\ U}

andX, U = {F exp X | F U = }

with U running over all open subsets of X.Observe that the sets of the second type form a subbase of the lower topol-

ogy l on exp X, while a subbase which consists of the sets of the first formdetermines a topology that is equal or weaker than the upper topology u onexp X. We call it an upper separation topology (not only for Tychonoff spaces)and denote by us. Thus the topology introduced in the latter lemma is a lowestupper bound of l and us. From now on we always consider exp X with thistopology, if otherwise is not specified. We also denote by exp l X, expu X and

expus X the set exp X with the respective topologies.If f : X Y is a continuous map of Tychonoff spaces, then we define themap exp f : exp X exp Y by the formula exp f(F) = Cl f(F). The equalityeexpY exp f = exp f eexpX implies that exp f is continuous, and we obtainan extension of the functor exp in Comp to Tych. Unfortunately, the extendedfunctor exp does not preserve embeddings.

Now we consider how to define valid inclusion hyperspaces in Tychonoffspaces.

Lemma 2.2. Let a family F of non-empty closed sets of a Tychonoff space Xis such that A B

clX, A F imply B F. Then the following properties

are equivalent :(a) F is a compact set in exp l X;

(b) for each monotonically decreasing net (F) of elements of F the inter-section

F also is in F.

Each such F is closed in expus X, hence in exp X. If X is compact, then theseconditions are also equivalent to :

(c) F is an inclusion hyperspace.

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Proof. Assume (a), and let (F) be a monotonically decreasing net of elementsofF. If

F / F, then the collection {X, X\ F} is an open cover of F that

does not contain a finite subcover, which contradicts the compactness of F inthe lower topology. Thus (a) implies (b).

Let (b) hold, and we have a cover ofF by subbase elements X, U, A.If there is no finite subcover, then F contains all sets of the form X \ (U1) Uk , 1, . . . , k A. These sets form a filtered family, which may beconsidered as a monotonically decreasing net of elements of F. Hence, by theassumption, F contains their non-empty intersection B = X\

A

U that does

not intersect any of U. This contradiction shows that each open cover of Fby subbase elements contains a finite subcover, and by Alexander Lemma F iscompact, i.e. (a) is valid.

Let F satisfy (b), and let C be a point of closure of F in expus X. Then foreach neighborhood U C there is F F such that F is completely separatedfrom X \ U, therefore Cl U F. The set U of all closures Cl U, with U aneighborhood of C, is filtered. Therefore

U = C F, hence F is closed

in expus X. If X is a compactum, then F satisfies the definition of inclusionhyperspace, i.e. (c) is true.

It is also obvious that an inclusion hyperspace on a compactum satisfies(b).

Therefore we call a collection F of non-empty closed sets of a Tychonoffspace X a compact inclusion hyperspace in X if A B

clX, A F imply

B F, and F is compact in the lower topology on exp X. Note that thelower topology is non-Hausdorff for non-degenerate X. The set of all compactinclusion hyperspaces in X will be denoted by GX.

Let G

X be the set of all inclusion hyperspaces G in X with the property :if A, B cl

X, A X = B X, then A G B G. Observe that each

such G does not contain subsets of X \ X.The latter lemma implies :

Proposition 2.3. A collection F exp X is a compact inclusion hyperspace ifand only if it is equal to {G X | G G} for a unique G GX .

We denote the map GX GX which sends each F GX to the respectiveG by eGX. It is easy to see that eGX(F) is equal to {G exp X | G X F }.

We define a Tychonoff topology on GX by the requirement that eGX is anembedding into GX. An obvious inclusion Gf(GX) GY for a continuousmap f : X Y allows to define a continuous map Gf : GX GY as arestriction of the map Gf, i.e. by the equality Gf eGX = eGY Gf. Ofcourse, Gf(F) = {G

clY | G f(F) for some F F} for F GX. A

functor G in the category of Tychonoff spaces is obtained. Its definition impliesthat eG = (eGX)XObTych is a natural transformation G U G, with allcomponents being embeddings, therefore G is a subfunctor of U G. Note alsothat eGX = GiX for all Tychonoff spaces X.

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Due to the form of the standard subbase of GX, we obtain :

Proposition 2.4. The topology on

GX can be determined by a subbase whichconsists of all sets of the form

U+ = {F GX | there is F F, F is completely separated from X\ U},

U = {F GX | F U = for all F F },

with U open in X.

Observe that this interpretation ofU+, U for Tychonoff spaces agrees withthe standard one for compact Hausdorff spaces.

As it was said before, the functor exp : Tych Tych does not preserveembeddings, thus we cannot regard exp expX as a subspace of expexp X,although exp X is a subspace of exp X. We can only say that image under expof the embedding exp X exp X is continuous. Therefore a straightforward

attempt to embed GX into exp2 X fails, while GX is embedded into exp2 X.Now we will show that the topology on GX is the weak topology with respect

to a collection of maps into the unit interval.

Lemma 2.5. Let a map : X I be continuous. Then the map : exp X Iwhich sends each non-empty closed subset F X to sup

xF(x) (or inf

xF(x)) is

continuous.

Proof. We prove for sup, the other case is analogous. Let supxF

(x) = < ,

, I. The set U = 1([0; +2 )) is open, and F is completely separated

from X \ U, hence F U. If G exp X, G U, then supxF

(x) +2 <

as well, and the preimage of the set [0; ) under the map is open.

Now let supxF(x) = > , , I. There exists a point x F such that

(x) > +2 , hence F intersects the open set U = 1((+2 ; 1]). Then X, U

F, and G exp X, G X, U implies supxG

(x) +2 > . Therefore the

preimage 1(; 1] is open as well, which implies the continuity of .

Lemma 2.6. Let a function : exp X I be continuous and monotonic. Then attains its minimal value on each compact inclusion hyperspace F GX.

Proof. If is continuous and monotonic, then it is lower semicontinuous withrespect to the lower topology. Then the image of the compact set F under iscompact in the topology {I (a, +) | a R} on I, therefore (F) contains aleast element.

Proposition 2.7. The topology on GX is the weakest among topologies suchthat for each continuous function : X I the map m which sends eachF GX to min{sup

F

| F F } is continuous. If : exp X I is a continuous

monotonic map, then the map which sends each F GX to min{(F) | F F}is continuous w.r.t. this topology.

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Proof. Let : exp X I be a continuous monotonic map, and min{(F) | F F} < , then there is F F such that (F) < . Due to continuity there is

a neighborhood U1, . . . , U k F such that (G) < for all G U1, . . . , U k.For is monotonic, the inequality (G) < is valid for all G U1 Uk.Therefore min{(G) | G G} < for all G (U1 Uk)

+, and the latteropen set contains F.

If min{(F) | F F} > , then (F) > for all F F. The function is continuous, hence each F F is in a basic neighborhood U0, U1, . . . , U k inexp X such that for all G in this neighborhood the inequality (G) > holds.We can assume that U1 U2 Uk is completely separated from X \ U0,then (G) > also for all G X, U1, U2, . . . , U k. The latter set is an openneighborhood of F in the lower topology. The set F is compact in exp l X,therefore we can choose a finite subcover U11 , . . . , U

1k1

, . . . , Un1 , . . . , U nkn

of F

such that G Ul1, . . . , U lkl

, 1 l n, implies (G) > . Then F is in an

open neighborhoodU =

{(U1j1 U

2j2

Unjn) | 1 j1 k1, 2 j2 k2, . . . n jn kn}.

Each element G of any compact inclusion hyperspace G U intersects allUl1, . . . , U

lkl

for at least one l {1, . . . , n}, therefore min{(G) | G G} > for

all G U. Thus min{(F) | F F } is continuous w.r.t. F GX.Due to Lemma 2.5 it implies that the map m : GX IC(X,I), m(F) =

(m(F))C(X,I) for F GX, is continuous.Now let F U+ for U

opX, i.e. there is F F and a continuous function

: X I such that |F 0, |X\U = 1. Then m(F) < 1/2, and for any

G GX the inequality m(G) < 1/2 implies G U+.IfF U, U

op

X, then due to the compactness of F we can choose V op

X

such that F V, and there is a continuous map : X I such that |V = 1,|X\U = 0. Then m(F) = 1 > 1/2, and for each G GX the inequalitym(G) > 1/2 implies G U. Therefore the inverse to m is continuous onm(GX), thus the map m : GX IC(X,I) is an embedding, which completesthe proof.

Remark 2.8. It is obvious that the topology on GX can be equivalently definedas the weak topology w.r.t. the collection of maps m : GX I, m(F) =max{inf

F | F F }, for all C(X, I).

Further we will need the subspace

GX = {F GX | for all F F there is a compactum K F, K F} GX.

It is easy to see that its image under eGX : GX GX is the set

GX = {G GX | for all G G there is a compactum K G X, K G},

and Gf(GX) GY for each continuous map f : X Y of Tychonoff spaces.Thus we obtain a subfunctor G of the functor G : Tych Tych.

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Lemma 2.9. Let X be a Tychonoff space. Then GX GeGX(G2X) eGX(GX).

The composition in the above inclusion is legal because GX = GX.

Proof. Let F G2X, F = GX GeGX(F), and F, G cl

X are such that

F X = G X. Assume F F, then there is H F such that F Gfor all G ClGX eGX(H), therefore for all G eGX(H). It is equivalent toF X H for all H H GX, which in particular implies that F X = .By the assumption, G X H for all H H as well, hence G G for allG eGX(H). The set of all H GX such that H A is closed for anyA exp X, thus G G for all G ClGX eGX(H). We infer that G F, andF GX.

For eGX is an embedding, we define GX as a map G2X GX such that

eG

X

G

X = G

X

GeG

X. This map is unique and continuous. Followingthe latter proof, we can see that

G(F) = {F exp X | F

H for some H F}, F G2X,

i.e. the formula is the same as in Comp.For the inclusion GX iX(X) eGX(GX) is also true, there is a unique

map GX : X GX such that eGX GX = GX iX, namely GX(x) ={F exp X | F x} for each x X, and this map is continuous. It isstraightforward to prove that the collections G = (GX)XObTych and G =(GX)XObTych are natural transformations resp. 1Tych G and G

2 G.

Theorem 2.10. The triple G = (G, G, G) is a monad in Tych.

Proof. Let X be a Tychonoff space and iX its embedding into X. Then :

eGX X GX = X GeGX GX =

X GX eGX = 1GX eGX = eGX,

thus GX GX = GX GGX = 1GX , similarly we obtain the equalitiesGX GGX = 1GX and GX GGX = GX GGX.

For GX, GX, GX coincide with GX, GX, GX for any compactum X,the monad G is an extension of the monad G in Comp to Tych.

3. Functional representation of the capacity monad in the categoryof compacta

In the sequel X is a compactum, c is a capacity on X and : X R is acontinuous function. We define the Sugeno integral of with respect to c bythe formula [10] :

X

(x) dc(x) = sup{c({x X | (x) }) | I}.

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The following theorem was recently obtained (in an equivalent form) byRadul [12] under more restrictive conditions, namely restrictions of normalized-

ness and non-expandability were also imposed. Therefore for the readers con-venience we provide a formulation and a short proof of a version more suitablefor our needs.

Theorem 3.1. LetX be a compactum, c a capacity on X. Then the functional

i : C(X, I) I, i() =

X

(x) dc(x) for C(X, I), has the following

properties :

(1) for all , C(X, I) the inequality (i.e. (x) (x) for allx X) implies i() i() (i is monotonic);

(2) i satisfies the equalities i( ) = i(), i( ) = i() for any I, C(X, I).

Conversely, any functional i : C(X, I) I satisfying (1),(2) has the form

i() =

X

(x) dc(x) for a uniquely determined capacity c MX.

In the two following lemmata i : C(X, I) I is a functional that satisfies(1),(2).

Lemma 3.2. If I and continuous functions , : X I are such that{x X | (x) } {x X | (x) } and i() , then i() .

Proof. For = 0 the statement is trivial. Otherwise assume i() . Let0 < . For , are continuous, there is (; ) such that the closedsets F = 1([0; ]) and G = 1([, 1]) have an empty intersection. Then, byBrouwer-Tietze-Urysohn Theorem, there is a continuous function : X [; ]such that |F , |G . Then we define a function f : X I as follows :

f(x) =

(x), x F,

(x), x / F G,

(x), x G.

Then f = , thus

i(f) = i( f) = i( ) = i() = ,

and i(f) = . Taking into account f = , we obtain

= i(f) = i( f) = i( ) = i(),

thus i() for all < . It implies i() .

Obviously if {x X | (x) } = {x X | (x) }, then i() ifand only if i() .

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Lemma 3.3. For each closed set F X and I the equality

inf{i() | F} = inf{i() | F}

is valid.

Proof. It is sufficient to observe that for all 0 < the sets { | F} and { | F} coincide, therefore by the previous lemma :

inf{i() | F} = inf{i() | F} = inf{i() | F}.

For the both expressions inf{i() | F} and inf{i() | F} donot exceed , they are equal.

Proof of the theorem. It is obvious that Sugeno integral w.r.t. a capacity sat-isfies (1),(2). If i is Sugeno integral w.r.t. some capacity c, then the equality

c(F) = inf{i() | F} must hold for all F cl X. To prove the converse, weassume that i : C(X, I) I satisfies (1),(2) and use the latter formula to definea set function c. It is obvious that the first two conditions of the definition ofcapacity hold for c. To show upper semicontinuity, assume that c(F) < forsome F

clX, I. Then there is a continuous function : X I such that

F, i() < . Let i() < < , then

i() = i() = i( ) c({x X | (x) }),

which implies c({x X | (x) }) < < . The set U = {x X | (x) > }is an open neighborhood of F such that c(G) < for all G

clX, G U. Thus

c is upper semicontinuous and therefore it is a capacity.The two previous lemmata imply that for any C(X, I) we have

i() = sup{ I | i() } = sup{ I | c({x X | (x) }) } =

sup{ c({x X | (x) }) | I} =

X

(x) dc(x).

Lemma 3.4. Let : X I be a continuous function. Then the map :

MX I which sends each capacity c to

X

(x) dc(x) is continuous.

Proof. Observe that

1(([0; )) = O(

1([0; ]), ), 1(((; 1]) = O+(

1((;1]), )

for all I.

Corollary 3.5. The map X IC(X,I) which sends each capacity c on X to((c))C(X,I) is an embedding.

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Recall that its image consists of all monotonic functionals from C(X, I) toI which satisfy (1),(2). Therefore from now on we identify each capacity and

the respective functional. By the latter statement the topology on M X can beequivalently defined as weak topology using Sugeno integral instead of Choquet

integral. We also write c() for

X

(x) dc(x).

The following observation is a trivial continuous version of Theorem 6.5 [10].

Proposition 3.6. LetC M X and C(X, I). Then MX(C)() = C().

Proof. Indeed, the both sides are greater or equal than I if and only ifC{c M X | c() } .

It is also easy to see that MX(x)() = (x) for all x X, C(X, I).Thus we have obtained a description of the capacity monad M in terms of

functionals which is a complete analogue of the description of the probabilitymonad P [5, 15]. Now we can easily reprove the continuity of MX and MX,as well as the fact that M = (M, M, M) is a monad.

4. Extensions of the capacity functor and the capacity monad to thecategory of Tychonoff spaces

We will extend the definition of capacity to Tychonoff spaces. A functionc : exp X {} I is called a regular capacity on a Tychonoff space X if itis monotonic, satisfies c() = 0, c(X) = 1 and the following property of uppersemicontinuity or outer regularity : if F

clX and c(F) < , I, then there

is an open set U F in X such that F and X \ U are completely separated,

and c

(G) < for all G U, G cl X.This definition implies that each closed set F is contained in some zero-set

Z such that c(F) = c(Z).Each capacity c on any compact space Y satisfies also the property which

is called -smoothness for additive measures and have two slightly differentformulations [1, 16]. Below we show that they are equivalent for Tychonoffspaces.

Lemma 4.1. LetX be a Tychonoff space and m : exp X{} I a monotonicfunction. Then the two following statements are equivalent :

(a) for each monotonically decreasing net (F) of closed sets in X and aclosed set G X, such that

F G, the inequality inf

c(F) c(G) is valid;

(b) for each monotonically decreasing net (Z) of zero-sets inX and a closedset G X, such that

Z G, the inequality inf

c(Z) c(G) is valid.

Proof. It is obvious that (a) implies (b). Let (b) hold, and let a net ( F) and aset G satisfy the conditions of (a). We denote the set of all pairs (F, a) suchthat a X \ F by A, and let be the set of all non-empty finite subsets

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of A. The space X is Tychonoff, hence for each pair (F, a) A there is azero-set Z,a F such that Z,a a. For = {(1, a1), . . . , (k, ak)} we

put Z = Z1,a1 Zk,ak . If is ordered by inclusion, then (Z) is amonotonically decreasing net such that

Z =

F G, thus inf

c(F)

inf

Z c(G), and (a) is valid.

We call a function c : exp X I a -smooth capacity if it is monotonic,satisfies c() = 0, c(X) = 1 and any of the two given above equivalent propertiesof-smoothness. It is obvious that each -smooth capacity is a regular capacity,but the converse is false. E.g. the function c : expN {} I which is definedby the formulae c() = 0, c(F) = 1 as F N, F = , is a regular capacity thatis not -smooth. For compacta the two classes coincide.

From now all capacities are -smooth, if otherwise is not specified.Now we show that capacities on a Tychonoff space X can be naturally identi-

fied with capacities with a certain property on the Stone-Cech compactificationX.

Lemma 4.2. Let c be a capacity on X. Then the following statements areequivalent :

(1) for each closed sets F, G X such that F X G, the inequalityc(F) c(G) is valid;

(2) for each monotonically decreasing net() of continuous functions X I and a continuous function : X I such that inf

(x) (x) for all

x X, the inequality inf

c() c() is valid.

Proof. (1) = (2). Let c() < , I, then c(Z0) < for the closed setZ0

={

x

X|

(x) }

. The intersection Z of the closed sets Z

={

xX | (x) } satisfies the inclusion ZX Z0, hence by (1): c(Z) c(Z0).

Due to -smoothness of c we obtain inf

c(Z) c(Z). Therefore there exists

an index such that c({x X | (x) }) < , thus c() < , andinf

c() < , which implies the required inequality.

(2) = (1). Let a continuous function : X I be such that |G = 1.Denote the set of all continuous functions : X I such that |F 1 by F.We consider the order on F which is reverse to natural: if , thenthe collection F can be regarded as a monotonically decreasing net such that((x))F converges to 1 for all x XG, and to 0 for all x X\ G. ThereforeinfF

(x) (x) for all x X, hence, by the assumption: infF

c() c().

Thus

inf{c() | : X I is continuous, |F 1}

inf{c() | : X I is continuous, |G 1},

i.e. c(F) c(G).

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We define the set of all c M X that satisfy (1) (2) by MX.Condition (1) implies that, if closed sets F, G X are such that F X =

G X, then c(F) = c(G). Therefore we can define a set function c : exp X {} I as follows : ifA

clX, then c(A) = c(F) for any set F

clX such that

F X = A. Obviously c(A) = inf{c() | C(X,I), A}.The following observation, although almost obvious, is a crucial point in our

exposition.

Proposition 4.3. A set function c : exp X {} I is equal to c for somec MX if and only if c is a -smooth capacity on X.

Therefore we define the set of all capacities on X by MX and identify it withthe subset MX MX. We obtain an injective map eMX : M X MX,and from now on we assume that a topology on MX is such that eMX is anembedding. Thus M X for a TychonoffX is Tychonoff as well.

If c is a capacity on X and : X I is a continuous function, we definethe Sugeno integral of w.r.t. c by the usual formula :

c() =

X

(x) dc(x) = sup{ c{x X | (x) } | I}.

For any continuous function : X I we denote by its Stone-Cechcompactification, i.e. its unique continuous extension to a function X I.

Proposition 4.4. Let c MX and c is defined as above. Then for anycontinuous function : X I we have c() = c().

Proof. It is sufficient to observe that

c({x X | (x) }) = c({x X | (x) }).

Thus the topology on MX can be equivalently defined as the weak-topologyusing Sugeno integral. It also immediately implies that the following theoremis valid.

Theorem 4.5. Let X be a Tychonoff space, c a capacity on X. Then the

functional i : C(X, I) I, i() =

X

(x) dc(x) for C(X, I), has the

following properties :

(1) for all , C(X, I) the inequality (i.e. (x) (x) for allx X) implies i() i() (i is monotonic);

(2) i satisfies the equalities i( ) = i(), i( ) = i() for any I, C(X, I);

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(3) for each monotonically decreasing net() of continuous functions X Iand a continuous function : X I such that inf

(x) (x) for all

x X, the inequality inf

i() i() is valid.

Conversely, any functional i : C(X, I) I satisfying (1)(3) has the form

i() =

X

(x) dc(x) for a uniquely determined capacity c MX.

Condition (3) is superfluous for a compact space X, but cannot be omittedfor noncompact spaces. E.g. the functional, which sends each C(R, I) tosup , has properties (1),(2), but fails to satisfy (3).

The following statement is an immediate corollary of an analogous theoremfor the compact case.

Proposition 4.6. The topology on MX can be equivalently determined by asubbase which consists of all sets of the form

O+(U, ) = {c MX | there is F cl

X,

F is completely separated from X\ U, c(F) > }

for all open U X, I, and of the form

O(F, ) = {c M X | c(F) < }

for all closed F X, I.

Like the compact case, for a continuous map f : X Y of Tychonoff spaceswe define a map Mf : MX MY by the two following equivalent formulae:Mf(c)(F) = c(f1(F)), with c MX, F

clY (if set functions are used),

or Mf(c)() = c( f) for c M X, C(X, I) (if we regard capacitiesas functionals). The latter representation implies the continuity of Mf, andwe obtain a functor M in the category Tych of Tychonoff spaces that is anextension of the capacity functor M in Comp.

The map eMX : MX M X coincides with MiX, where iX is the em-bedding X X (we identify MX and M X), and the collection eM =(eMX)XOb Tych is a natural transformation from the functor M to the func-tor U M, with U : Comp Tych being the inclusion functor. Observe thatMX(X) MX = eMX(MX), therefore there is a continuous restrictionMX = MX|X : X M X which is a component of a natural transforma-tion eM : 1Tych M. For all x X Ob Tych, F

cl

X the value GX(x)(F)

is equal to 1 if x F, otherwise is equal to 0.

Lemma 4.7. Let X be a Tychonoff space. Then MX MeMX(M2X) eMX(MX).

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Proof. Let C M2X, and F, G cl

X are such that F X G. Then for all

c M

X we have c(F)

c(G), thus for each I :{c MX | eMX(c)(F) } {c M X | eMX(c)(G) },

hence

MeMX(C)({c M X | c(F) })) = C(eMX1({c M X | c(F) }))

C(eMX1({c MX | c(G) })) = M eMX(C)({c MX | c(G) })),

thus

MX M eMX(C)(F) = sup{ MeMX(C)({c M X | c(F) })

sup{ MeMX(C)({c MX | c(G) }) = MX M eMX(C)(G),

which means that M

X MeM

X(C) MX = eM

X(MX).

For eMX : MX MX is an embedding, there is a unique map MX :M2X M X such that MXMeMX = eMXMX, and this map is contin-uous. It is straightforward to verify that the collection M = (MX)XObTychis a natural transformation M2 M, and MX can be defined directly, withoutinvolving Stone-Cech compactifications, by the usual formulae :

MX(C)(F) = sup{ C({c MX | c(F) }), C M2X, F

clX,

or

MX(C)() = C(), C(X, I), where (c) = c() for all c MX.

Theorem 4.8. The triple M = (M, M, M) is a monad in Tych.

Proof is a complete analogue of the proof of Proposition 2.10.This monad is an extension of the monad M = (M, M, M) in Comp in

the sense that M X = MX, MX = MX and MX = MX for each com-pactum X.

Proposition 4.9. Let for each compact inclusion hyperspace F on a Tychonoffspace X the set function iMG X(F) : exp X {} I be defined by the formula

iMG X(F)(A) =

1, A F,

0, A / F,A

clX.

TheniKGX is an embeddingGX M X, and the collectioniKG = (i

KGX)XObTych

is a morphism of monads G M.

Thus the monad G is a submonad of the monad M.Now let

MX = {c MX | c(A) = sup{c(F) | F AX is compact} for all A cl

X}.

It is easy to see that MX MX. As a corollary we obtain

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Proposition 4.10. A set function c : exp X {} I is equal to c forsome c MX if and only if c is a -smooth capacity on X and satisfies the

condition c(A) = sup{c(F) | F A is compact} for allA cl

X (inner compact

regularity).

If a set function satisfies (1)(4), we call it a Radon capacity. The set of allRadon capacities on X is denoted by M X and regarded as a subspace of M X.An obvious inclusion M f(MX) MY for a continuous map f : X Y ofTychonoff spaces implies Mf(M X) MY. Therefore we denote the restrictionof M f to a mapping MX MY by Mf and obtain a subfunctor M of thefunctor M.

Question 4.11. What are necessary and sufficient conditions for a functional

i : C(X, I) I to have the form i() =

X

(x) dc(x) for a some capacity

c MX?

Here is a necessary condition : for each monotonically increasing net ()of continuous functions X I and a continuous function : X I such thatsup

(x) (x) for all x X, the inequality sup

i() i() is valid.

The problem of existence of a restriction of MX to a map M2X MX is

still unsolved and is connected with a similar question for inclusion hyperspacesby the following

Proposition 4.12. Let X be a Tychonoff space. If MX(M2X) MX, thenGX(G2X) GX.

Proof. We will consider equivalent inclusions MX(M2X) MXand GX(G

2X)

GX. The latter one means that, for each set A cl X and compact set G GXsuch that each element F of any inclusion hyperspace B G contains a com-pactum K B , K X, there is a compact set H A, H

G.

Assume that GX(G2X) GX, then there are A cl

X and a compact

set G GX such that all inclusion hyperspaces in G contain subsets of A, butthere are no compact subsets of A in

G. For each B G let a capacity cB be

defined as follows :

cB(F) =

1, F B ,

0, F / B ,F

clX.

It is obvious that cB MX, and the correspondence B cB is continuous,

thus the set B = {cB | B G} MX is compact. Therefore the capacityC M2X, defined as

C(F) =

1, F B,

0, F B,F

clMX,

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is in M(MX). Then MX(C)(ClX A) = 1, but there is no compact subsetK A such that cB(K) = 0 for all B G, therefore MX(C)(K) = 0 for all

compact K A = ClX A X, and MX(C) / MX.

It is still unknown to the authors :

Question 4.13. Does the converse implication hold? Do all locally compactHausdorff or (complete) metrizable spaces satisfy the condition of the previousstatement?

5. Topological properties of the functors G, G, M and M

Recall that a continuous map of topological spaces is proper if the preimageof each compact set under it is compact. A perfect map is a closed continuousmap such that the preimage of each point is compact. Any perfect map is

proper [4].From now on all maps in this section are considered continuous, and allspaces are Tychonoff if otherwise not specified.

Remark 5.1. We have already seen that properties of the functors M and Mare parallel to properties of the functors G and G. Therefore in this sectionwe present only formulations and proofs of statements for M and M. All ofthem are valid also for G and G, and it is an easy exercise to simplify the proofsfor capacities to obtain proofs for compact inclusion hyperspaces.

Proposition 5.2. Functors M and M preserves the class of injective maps.

Proof. Let f : X Y be injective. If c, c MX and A cl

X are such that

c(A) = c(A), then B = Cl f(A)

cl

Y, and Mf(c)(B) = c(f1(B)) = c(A) =

c(A) = c(f1(B)) = M f(c)(B), hence Mf(c) = M f(c), and M f is injective,as well as its restriction M f.

Proposition 5.3. Functors M and M preserve the class of closed embeddings.

Proof. Let a map f : X Y be a closed embedding (thus a perfect map), thenfor the Stone-Cech compactification f : X Y the inclusion f(X\X) Y \ Y is valid [4]. We know that MX(MX) MY, M X(MX) MY.

Let c MX \ MX, then there are F, G cl

M X such that F X G,

but c(F) > c(G). Then f(F) and f(G) are closed in Y, and f(F) \ f(G) f(X \ X) Y \ Y.

The sets F = f1(f(F)) and G = f1(f(G)) are closed in X and satisfyF X = F X, G X = G X, thus c(F) = Mf(c)(f(F)) > c(G) =Mf(c)(f(G)), which implies Mf(c) / MY. Thus (M f)1(MY) = MX,and the restriction M f|MX : MX MY is perfect, therefore closed. It isobvious that this restriction is injective, thus is an embedding. For the mapsMf|MX and Mf are homeomorphic, the same holds for the latter map.

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Now let c M X\ MX, i.e. there is F cl

X such that c(F) > sup{c(K) |

K F X is compact}. The compact set F

= f(F) is closed in Cl f(X) Y. Observe that F = (f)1(F) and obtain:

sup{Mf(c)(L) | L F Y is compact} =

sup{c((f)1(L)) | L F Y is compact}

sup{c(K) | K F X is compact} < c(F) = Mf(c)(F),

and Mf(c) / MX. The rest of the proof is analogous to the previous case.

It allows for a closed subspace X0 X to identify M X0 and MX0 with theimages of the map M i and Mi, with i : X0 X being the embedding.

We say that a functor F in Tych preserves intersections (of closed sets) if forany space X and a family (i : X X) of (closed) embeddings the equality F X = F X0 holds, i.e.

F i(X) = F i0(X0), where i0 is the embeddingof X0 =

X into X. This notion is usually used for functors which preserve

(closed) embeddings, therefore we verify that:

Proposition 5.4. Functors M and M preserve intersections of closed sets.

Proof. Let c MX and closed subspaces X X, A, are such thatc MX for all A. Let 2

Af be the set of all non-empty finite subsets

of A. It is a directed poset when ordered by inclusion. For all F cl

X and

{1, . . . , k} 2Af we have c(F) = c(F X1 Xk). The monotonicallydecreasing net (F X1 Xk){1,...,k}2Af converges to F X0, with

X0 =

A X. Thus c(F) = c(F X0), which implies c MX0.

The statement for M is obtained as a corollary due to the following obser-

vation: if X0 X is a closed subspace, then MX0 = MX M X0.

Therefore for each element c MX there is a least closed subspace X0 Xsuch that c M X0. It is called the support of c and denoted supp c.

It is unknown to the author whether the functor M preserve finite or count-able intersections.

Proposition 5.5. Functor M preserves countable intersections.

Proof. Let c M X belong to all MXn for a sequence of subspaces Xn X,n = 1, 2, . . . . If A

clF, > 0, then there is a compactum K1 A X1

such that c(K1) > c(A) /2. Then choose a compactum K2 K1 X2such that c(K2) > c(K1) /4, . . . , a compactum Kn Kn1 Xn such that

c(Kn) > c(Kn1) /2n

, etc. The intersection K =n=1 Kn is a compact

subset of A X0, X0 =n=1 Xn, and c(K) > C(A) . Thus sup{c(K) | K

A X0 is compact} = c(A) for all A cl

X, i.e. c MX0.

It is easy to show that M and M do not preserve uncountable intersections.

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We say that a functor F in Tych (or in Comp) preserves preimages if foreach continuous map f : X Y and a closed subspace Y0 Y the inclusion

F f(b) F Y0 for b F X implies b F(f1(Y0)), or, more formally, F f(b) F j(F Y0) implies b F i(F(f1(Y0))), where i : f1(Y0) X and j : Y0 Yare the embeddings.

Proposition 5.6. Functors M and M do not preserve preimages.

It is sufficient to recall that the capacity functor M : Comp Comp, beingthe restriction of the two functors in question, does not preserve preimages [18].

Proposition 5.7. Letf : X Y is a continuous map such that f(X) is densein Y. Then Mf(MX) is dense in MY, and Mf(MX) is dense in MY.

Proof. Let MX be the set of all capacities on X with finite support, i.e.

MX =

{M K | K X is finite}

Then MX M X MX, M f(MX) = M(f(X)), and the latter set isdense in both M Y and MY.

6. Subgraphs of capacities on Tychonoff space and fuzzy integrals

In [18] for each capacity c on a compactum X its subgraph was defined asfollows :

sub c = {(F, ) exp X I | c(F)}.

Given the subgraph sub c, each capacity c is uniquely restored : c(F) = max{ I | (F, ) sub c} for each F exp X.

Moreover, the map sub is an embedding M X exp(exp X I). Its imageconsists of all sets S exp X I such that [18] the following conditions aresatisfied for all closed nonempty subsets F, G of X and all , I :

(1) if (F, ) S, , then (F, ) S;

(2) if (F, ), (G, ) S, then (F G, ) S;

(3) S exp X {0} {X} I;

(4) S is closed.

The topology on the subspace sub(M X) exp(exp X I) can be equivalentlydetermined by the subbase which consists of all sets of the form

V+(U, ) = {S sub(MX) | there is (F, ) S, F U, > }

for all open U X, I, and of the form

V(F, ) = {S sub(MX) | < for all (F, ) S}

for all closed F X, I.Let the subgraph of a -smooth capacity c on a Tychonoff space X be defined

by the same formula at the beginning of the section. Consider the intersection

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sub c (exp X {}). It is equal to S(c) {}, with S(c) = {F exp X |c(F) }. The latter set is called the -section [18] of the capacity c and is

a compact inclusion hyperspace for each > 0. Of course, S0(c) = exp X isnot compact if X is not compact. If 0 < 1, then S(c) S, andS(c) =

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Inclusion hyperspaces and capacities on Tychonoﬀ spaces: functors and monads