Topological properties preserved by scatteredly continuous maps
by
Bogdan Bokalo
Ivan Franko National University of Lviv
Coauthors: Taras Banakh

In the talk we detect topological properties preserved by scatteredly continuous maps. A map f:X→ Y between topological spaces is scatteredly continuous if for each non-empty subspace A ⊂ X the restriction f|A has a point of continuity. A bijective map f:X→ Y is called a scattered homeomorphism if f and f^-1 are scatteredly continuous.

We shall say that a class P of (regular) topological spaces is

• topological if a space X belongs to P provided X is homeomorphic to a space Y ∈ P;

• closed-hereditary (resp. open-hereditary) if for each space X ∈ P, every closed (resp. open) subspace Y of X belongs to P;

• scatteredly projective if a (regular) space Y belongs to P provided Y is the image of a space X ∈ P under a (scatteredly) continuous map f:X→ Y;

• s-additive if a (regular) space X belongs to P if it has a countable closed cover C ⊂ P.

By the additivity add(P) of a class P of (regular) spaces we understand the cardinal k for which there is a (regular) space X ∉ P that has a closed cover C ⊂ P of size |C| ≤ k. If no such a cardinal k exists, then we put add(P)=∞ and assume that k < ∞ for all cardinals k. Observe that P is s-additive if and only if add(P) > ℵ₀. By definition, for a topological T₁-space X the large pseudocharacter Y(X) is equal to the smallest cardinal k such that each closed subset F ⊂ X can be written as the intersection ∩U of a family U consisting ≤ k open subsets of X. It is easy to see that Y(X) ≤ hl(X) for every regular space X. By the paracompactness number par(X) of a topological space X we understand the smallest cardinal k such that each open cover of X can be refined by an closed cover F of X that can be written as the union F =∪_{a < k}F_a of k many locally finite families F_a of closed subsets of X. Hence a topological space X is paracompact if and only if par(X) ≤ 1

Proposition 1 Let P be a closed-hereditary projective class of topological spaces and f:X→ Y be a surjective scatteredly continuous map from a space X ∈ P onto a regular topological space Y. If max{par(X), Y(X)} < add(P), then Y ∈ P.

Corollary 2 Let f:X→ Y be a scatteredly continuous surjective map between regular spaces. If the space X is hereditarily Lindelöf and s-compact, then so is the space Y.

Corollary 3 A regular space X is analytic if and only if it is the image of a Polish space P under a scatteredly continuous map f:P→ X.

Now we detect some cardinal functions that respect scatteredly continuous maps. We define a cardinal function j on a closed-hereditary class T of topological spaces to be

• topological-invariant if j(X)=j(Y) for any homeomorphic spaces X, Y ∈ T;

• additive if j(X) ≤ ∑_{C ∈ C}j(C) for any closed cover C of a space X ∈ T;

• closed-hereditary (resp. open hereditary) if j(Y) ≤ j(X) for every closed (resp. open) subspace Y of a space X ∈ T;

• scatteredly projective if j(f(X)) ≤ j(X) for every scatteredly continuous map f:X→ Y between spaces X, Y ∈ T;

• global if j(D) ≥ |D| for any discrete space D ∈ T.

Corollary 4 Let j be an additive, projective, and closed-hereditary cardinal function on the class of regular spaces. For any surjective scatteredly continuous map f:X→ Y between regular spaces we get j(Y) ≤ max{j(X), par(X), Y(X)} ≤ max{j(X), hl(X)}.

Theorem 5 A global additive closed-hereditary cardinal function j on the class of regular spaces is scatteredly projective if and only if j is projective and j ≥ hl.

Corollary 6 If f:X→ Y is a scatteredly continuous map between regular spaces, then

1. nw(Y) ≤ nw(X);

2. hl(Y) ≤ hl(X);

3. hd(Y) ≤ max{hd(X), hl(X)}.

In fact, the second item of Corollary 6 holds for non-regular spaces too.

Because of an example of a regular space X with hd(X) < hl(X) , Theorem 5 implies that the hereditary density hd is not scatteredly-projective, which means that there is a scatteredly continuous map f:X→ Y between regular spaces such that hd(Y) > hd(X).

Example 7 Under the Diamond-Axiom A.Ostaszewski has constructed a regular space X which is uncountable, compact, scattered, and hereditarily separable. Then any bijective map f:X→ D to a discrete space D is scatteredly continuous but hd(D)=|D|=|X| > ℵ₀=hd(X). This yields that the class of regular hereditarily separable spaces is not scatteredly projective under the Diamond-Axiom.

On the other hand, S.Todorcevic has constructed a model of ZFC without S-spaces, that is, regular hereditarily separable non-Lindelöf spaces. In such models the class of regular hereditarily separable spaces is scatteredly projective.

Next, we shall show that scatteredly homeomorphic spaces can be decomposed into a sum of closed homeomorphic subspaces.

Theorem 8 If h:X→ Y is a scattered homeomorphism between regular spaces, then there are closed covers {X_i:i ∈ I} and {Y_i:i ∈ I} of X and Y for some index set I of size |I| ≤ max{par(X), par(Y), Y(X), Y(Y)} such that for each i ∈ I the restriction h|X_i is a homeomorphism of X_i onto Y_i.

Theorem 8 can be partly reversed. Namely, for scattered homeomorphisms a theorem of Cantor-Bernstein type holds.

Theorem 9 Two topological spaces X, Y are scatteredly homeomorphic if each of then is homeomorphic to a closed subspace of the other space.

Remark 10 Remark that the Baire space N^w and the Cantor cube 2^w embed into each other, but fail to be scatteredly homeomorphic. The reason is that scattered homeomorphisms preserve the s-compactness, see Corollary 2. This shows that the closedness is essential in Theorem 9.

Theorem 11 Let P be a closed-hereditary topological property.

1. A regular space X with max{par(X), Y(X)} < add(P) has property P if and only if X is scatteredly homeomorphic to a regular space Y with property P and max{par(Y), Y(Y)} < add(P);

2. A regular space X has property P if and only if X is scatteredly homeomorphic to a regular space Y with property P and hl(Y) ≤ add(P).

Applying this theorem to s-additive topological properties, we get

Corollary 12 Let P be a closed-hereditary s-additive topological property of perfectly paracompact spaces. A perfectly paracompact space X has property P if and only if X is scatteredly homeomorphic to a perfectly paracompact space Y with property P.

Corollary 13 If X, Y are scatteredly homeomorphic perfectly paracompact spaces, then

1. nw(X)=nw(Y);

2. hd(X)=hd(Y);

3. dimX=dimY;

4. X is s-compact iff so is the space Y;

5. X is analytic iff so is the space Y.

Date received: May 12, 2008