There is more than one useful way to create a new ...settheory.mathtalks.org/wp-content/uploads/2012/06/lucia_junqueira.pdfIntroduction Theorem For any set H and X H , there is an

There is more than one useful way to create a new

topological space from a space and an elementary

submodel containing it

Lúcia R. Junqueira

Instituto de Matemática e Estatística

Universidade de São Paulo

[email protected]

5 de junho de 2012

Lúcia R. Junqueira (IME�USP) There is more than one useful way to create a new topological space... 1 / 32

Introduction

The use of elementary submodels has become a standard tool inset-theoretic topology and in�nitary combinatorics.

De�nition

M is an elementary submodel of N (denoted by M ≺ N) if M ⊆ N and forevery n ∈ ω and formula ϕ with at most n free variables and all{a1, ..., an} ⊆ M,

ϕM(a1, ..., an) holds i� ϕN(a1, ..., an) holds

(where ϕM is the formula obtained by restricting the quanti�ers to M).


Introduction

Theorem

For any set H and X ⊆ H, there is an elementary submodel M of H suchthat X ⊆ M and |M| ≤ |X |+ ℵ0.

De�nition

For a cardinal θ, Hθ is the set of all sets of hereditary cardinality < θ.

For θ regular, Hθ |= ZFC − P . We will always take θ su�ciently large sothat Hθ contains all sets of interest in the context under discussion.


Introduction

In the paper [JT], we directed our attention to the study of the tool itself,i.e., we investigated a new topological space de�ned using elementarysubmodels.

De�nition

Let 〈X , T 〉 be a topological space and let〈X , T 〉 ∈ M ≺ Hθ. We de�ne XM = X ∩M with the topology generatedby {U ∩M : U ∈ T ∩M}.


Introduction

Later, T. Eisworth used a di�erent kind of construction, done before byBandlow and Dow independently, to prove results about compactmonotonically normal spaces.

Instead of looking at a substructure, as we did in XM , they looked at aquotient. Using an elementary submodel M and a Tychono� space X , theyde�ned an equivalence relation and then looked at the quotient X/M, butwith a topology weaker than the quotient topology.


Introduction

De�nition

For x , y ∈ X , x ∼ y if and only if f (x) = f (y) for every continuousfunction f : X −→ R in M. Equivalently, we can de�ne x ∼ y if and only ifx ∈ V ↔ y ∈ V , for every co-zero set V ∈ M.

De�nition

Let X/M be the set of all the equivalence classes of points in X with thetopology genarated by {π(V ) : V ∈ M and V is a co-zero set }, where π isthe natural projection that takes x to [x ].

More recently P. Burton and F. Tall used this construction to prove resultsrelated to some important problems about Lindelöf spaces.


Introduction

We will do the "elementary quotient" in a more general way, show somebasic results and give some applications.

For the rest of this talk, M will always be such that 〈X , T 〉 ∈ M andM ≺ Hθ, where θ is large enough to make the argument at hand work.


A more general X/M

Let X be a topological space, B a basis for X and M a �xed elementarysubmodel with X ,B ∈ M. We de�ne the following equivalence relation onX :

De�nition

x ∼B y if and only if x ∈ V ↔ y ∈ V for every V ∈ B ∩M.

We will omit metioning B and just write ∼.

Let X/(M,B) be the set of all the equivalence classes of points in X and πthe natural projection map from X to X/(M,B).


A more general X/M

We want to de�ne the topology on X/(M,B) to be the one generated by

{π(V ) : V ∈ B ∩M}.

However, without extra conditions on B, this does not have to be a basefor a topology on X/(M,B).

For instance, take X to be a discrete space, M such that |M| < |X | andB = {{x} : x ∈ X}. Then if y /∈ M, there is no V ∈ B ∩M such that[y ] ∈ π(V ).

We then need to have extra assumptions on B. It is easy to show thefollowing:


A more general X/M

Proposition

If B is closed under �nite intersections and is such that X ∈ B , then{π(V ) : V ∈ B ∩M} is a basis for a topology on X/(M,B).

From now on we will assume B satis�es these two conditions andX/(M,B) will denote the topological space obtained.


A more general X/M

It is easy to see that:

Lemma

π is a continuous function.

For a Tychono� space, Eisworth de�nition of X/M is, in our notation,X/(M,B), where B is the set of all the co-zero sets of X .

In general we do not have nice preservation results. For instance, we canstart with X normal, even metrizable, but have that X/(M,B) is not T1.Like in the X/M case, we will restrict ourselves to Tychono� spaces.


A more general X/M

If X is a Tychono� space, then X can be embedded in [0, 1]C , where C is aset of functions from X to [0, 1] that separates points and closed sets. Theembedding i : X −→ [0, 1]C is de�ned by i(x) = (f (x))f ∈C .

From now on we will be always assuming that X is Tychono�, C is �xedand we will identify X with its homeomorphic image i [X ] and x ∈ X withi(x). Note that i ∈ M, so this identi�cation will not cause us problemwhen we do the elementary submodel arguments.

Let BI be a countable basis of [0, 1] with BI ∈ M, and therefore BI ⊆ M.


A more general X/M

De�nition

We say a base B for X satis�es (*) if B is a base for X induced by theusual product topology base of [0, 1]C where in each coordinate we onlypick open sets from BI .

De�ne the following equivalent relation:

x ∼0 y if and only if f (x) = f (y) for every f ∈ C ∩M.

Lemma

x ∼B y if and only if x ∼0 y .


A more general X/M

Theorem

If B satis�es (*), then X/(M,B) is homeomorphic to a subspace of[0, 1]C∩M . In particular, if X = [0, 1]C , then X/(M,B) = [0, 1]C∩M .

Corollary

X/(M,B) is Tychono�.

Corollary

If C = C(X ), then X/(M,B) = X/M

Corollary

The equivalence classes are closed subsets of X .


Examples

We give examples that will show that by choosing di�erent basis B ordi�erent set of functions C, we can get very di�erent spaces X/(M,B).

For all the examples bellow, �x X a discrete space of cardinality ω1 and Ma countable elementary submodel. In each of them we will choose adi�erent B or C.

Example

If B = {{x} : x ∈ X} ∪ {X}, then X/(M,B) is not even T1.

Example

If B = {A ⊆ X : A is �nite or co�nite }, then X/(M,B) is homeomorphicto a convergent sequence. This is equivalent to use C be the set of all thecaracteristic functions of the unitary sets.


Examples

Example

If B = {A ⊆ X : A is �nite or co�nite } ∪ {A,X \A}, for some A ∈ M suchthat both A and X \ A have cardinality ℵ1, then X/(M,B) ishomeomorphic to two convergent sequences.

Example

X/M is an uncountable separable metrizable space.

Proof

Since M is countable X/M has a countable base, so it is separablemetrizable. To show that X/M is uncountable, just note that {x} = [x ] forevery x ∈ X . Indeed, |X | = ℵ1, so there is f : X −→ R one-to-one. Byelementarity, we can get f ∈ M. Also, since X has the discrete topology, fis continuous. So f witnesses that no x and y can be equivalent.


Basic topological operations

Let X be a subspace of Y and �x B a base for Y . ThenB∗ = {V ∩ X : V ∈ B} is a base for X . Note that if B satis�es (*), thenB∗ also does.

Proposition

X/(M,B∗) is homeomorphic to a subspace of Y /(M,B).

Corollary

If X is a dense subspace of Y , then X/(M,B∗) is homeomorphic to adense subspace of Y /(M,B).



If Y is Tychono�, C is a set of continuous functions separating points andclosed sets and B satis�es (*), then we have:

Corollary

X/(M,B) = X/(M,B), where the �rst closure is taken in [0, 1]C∩M andthe closure of X is taken in [0, 1]C .

Note that for X/M, the closure of X will be βX . However, even thoughclosure of X/(M,B) is a compacti�cation, it may not be the Stone-�echone. For instance, if M is countable, it will always be a metrizable one.



We now look at the product.

For each i ∈ ω, let Xi ∈ M and �x a base Bi ∈ M for Xi , with Xi ∈ Bi andBi closed under �nite intersections.For the product X =

∏n∈ω Xn we consider the natural canonical base B

constructed from Bi , i ∈ ω. Note that since Xi , Bi ∈ M, for each i ∈ ω, wehave that

∏i∈ω Xi ∈ M and B ∈ M. Also, X ∈ B and B is closed under

�nite intersections.

Proposition∏i∈ω(Xi/(M,Bi ) is the same as (

∏i∈ω Xi )/(M,B).


Two results

The following results are due to T. Eisworth.

Proposition

X is pseudocompact Tychono� if and only if for every countableelementary submodel M the space X/M is compact.

Proof

If X is pseudocompact, then X/M is peseudocompact and therefore it iscompact (since it is metrizable).Suppose now that X is not pseudocompact. Then there is f : X −→ Rsuch that f [X ] is unbounded. By elementarity, we can pick f ∈ M. Butthen f induces f ∗ : X/M −→ R de�ned by f ∗([x ]) = f (x). Note that therange of f ∗ is the same as the range of f , a contradiction since X/M iscompact.


Two results

We can also give a caracterization of realcompactness, but for that we needto use a family of countable elementary submodels.Let Σ = {M ≺ Hλ : X ∈ M and |M| = ω}, ordered by ∈. If M ∈ N, thenthere is a natural map πN,M : X/N −→ X/M onto.

Theorem

X is realcompact if and only if X is homeomorphic to the inverse limit of{X/M : M ∈ Σ}.


Applications

De�nition

A Lindelöf space X is productively Lindelöf if X × Y is Lindelöf for everyLindelöf space Y . X is powerfully Lindelöf if its countable powers areLindelöf.

An old question of E.A. Michael asks:

Problem

Is every productively Lindelöf space powefully Lindelöf?

Alster showed:

Theorem

CH implies that every productively Lindelöf space of weight ≤ ℵ1 ispowerfully Lindelöf.


Applications

In [BT], P. Burton and F. Tall got a small improvement of this result usingX/M. They showed that the same result is true if we assume thatL(Xω) ≤ ℵ1.

The idea of the proof is to use Alster result for X/M, where M is anelementary submodel of size ℵ1. Note that in this case w(X/M) ≤ ℵ1.

The main di�cult is to get the preservation of Lindelöf from Xω/M to Xω.

Theorem

(CH) If X is regular productively Lindelöf and satis�es the preservationproperty above, then Xω is Lindelöf.


Applications

Demonstração.

Fix B a base for X and �x M an elementary submodel of size ℵ1 withX ,B ∈ M.Since X/(M,B) is a countinuous image of X , it is also productivelyLindelöf. Now, by the de�nition of the topology of X/(M,B), we have thatw(X/(M,B)) ≤ ℵ1. Since we are assuming CH, by Alter�s result,X/(M,B) is powerfully Lindelöf.We want to show that Xω is Lindelöf. For that, �x V a base for Xω suchthat the product result works.But then (Xω)/(M,V) = (X/(M,B))ω, which is Lindelöf. The result thenfollows from our preservation assumption.


Applications

We improved their result using the following caracterization of Lindelöfspaces:

Proposition

Let X be a Tychono� space. Then X is Lindelöf if and only if for some(every) compacti�cation bX of X , it is true that for any compact setF ⊆ bX \ X there is a closed Gδ-set P in bX such that F ⊆ P andP ∩ X = ∅.

For the next result we will, as before, identify the Tychono� space X withits homeomorphic copy in [0, 1]C , where C separate points and closed sets.We will look at the closure of X in [0, 1]C to get a compati�cation.


Applications

We then get the following preservation result:

Theorem

Let X be a Tychono� space, B a base of X satisfying (*) and M andelementary submodel of cardinality ω1, with everything needed in M. If Xis such that for every compact F ⊆ X \ X in M,π(F ) ⊆ X/(M,B) \ X/(M,B), then X/(M,B) Lindelöf implies that X isLindelöf.


Applications

Demonstração.

Suppose that X is not Lindelöf. Then there is F ⊆ X \ X contradicting thecondition of the previous preposition. By elementarity, we can take F ∈ M.By our assumption, we have thatπ(F ) ⊆ X/(M,B) \ X/(M,B) = X/(M,B) \ X/(M,B) (closures taken in[0, 1]C and [0, 1]C∩M respectively). Note that since F is compact, thenπ(F ) is also compact.Since X/(M,B) is Lindelöf, there is a closed Gδ set (in X/(M,B))K ⊆ X/(M,B) \ X/(M,B) such that π(F ) ⊆ K . It is now straightforwardto check that P = π−1(F ) satis�es the condition of the previousproposition, contradicting the choice of F .


Applications

Corollary

Suppose X and M as in the theorem and suppose that X also satis�es thatfor every compact F ⊆ X \ X there is D ⊆ B such that |D| ≤ ω1, D coversX and F ∩

⋃D = ∅. Then X/(M,B) Lindelöf implies that X is Lindelöf.

Theorem

(CH) If X is a regular productively Lindelöf and such that for everycompact F ⊆ Xω \ Xω there is D ⊆ B such that |D| ≤ ω1, D covers Xω

and F ∩⋃D = ∅, then Xω is Lindelöf.


Applications

It is easy to see that if L(X ) ≤ ℵ1 implies the condition above, so we have:

Corollary (BT)

(CH) If X is a regular productively Lindelöf and L(Xω) ≤ ℵ1, then Xω isLindelöf.


Applications

We �nish by mentioning another kind of application, done by F. Tall and T.Usuba. Other applications of this type was also obtained by P. Burtonindependently.

De�nition

A Lindelöf space is indestructible if it generates a Lindelöf topology in anycountable closed forcing extension.

F. Tall and Usuba showed:

Theorem

Let κ be an inaccessible cardinal and G a P-generic where P is theLévy-collapse making κ = ω2. Then in V [G ] every Lindelöf space of weight≤ ℵ1 and size > ℵ1 is destructible.


Applications

To prove this result they considered chains of elementary submodels. Theequivalence classes will produce deacrasing chains of closed sets which willcorrespond to branches of a certain tree.

Similar ideas was also used by P. Burton to prove results related to Lindelöfspaces.


Referências

[BT] P. Burton and F. D. Tall, Productive Lindelöfness and a class of spaces

considered by Z. Frolík, Topology and its Applications, to appear.

[E] T. Eisworth, Elementary submodels and separable monotonically

normal compacta, Topology Proceedings 30 (2006), 431�443.

[JT] L. R. Junqueira and F. D. Tall, The topology of elementarysubmodels, Topology and its Applications 82 (1998), 239�266.


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There is more than one useful way to create a new ...settheory.mathtalks.org/wp-content/uploads/2012/06/lucia_junqueira.pdfIntroduction Theorem For any set H and X H , there is an