# Intersections of open sets and $\alpha$-favorable spaces

I would like to ask about some classes of topological spaces whether they have been studied, what are they called (if they have a name) and whether they have some interesting properties. For the purpose of this post I have made up some names for these classes of spaces. Below I will write down what I was able to find out by myself and I will also try to give some motivation for the question.

I should mention that this is modification of very similar question as I have asked at MSE. (Although I hope that I have managed to make it a little more concise.)

Question 1. What is name/characterization of spaces such that $\bigcap \mathcal F\ne\emptyset$ for every open ultrafilter $\mathcal F$. Let us call such spaces u.i.-spaces. (Where "u.i." stands for ultrafilter intersection).1

Before asking about other similar classes of spaces, let me digress a little and mention my motivation for asking this. I will also mention some equivalent conditions to the above property.

These classes of spaces caught my attention after I have seen the following result2, which says that every u.i.-space is Baire:

Proposition. If there are no free open ultrafilters on a space $X$, then $X$ is a Baire space.

After seeing this result I asked myself what the spaces where this condition is fulfilled look like. (It seems a reasonable question to ask about spaces fulfilling the assumptions of a result you see published somewhere.)

Using more-or-less standard arguments we can find out that the following conditions are equivalent:

• Every closed cover of the space $X$ has a finite subcover.
• Every system of open subsets of $X$ which has finite intersection property has a non-empty intersection.
• Every open filter on $X$ has a non-empty intersection.
• Every open ultrafilter on $X$ has a non-empty intersection.
• The space $X$ has a finite dense subset. (This last equivalent condition is mentioned in [D, Theorem 3.3].)

I was able to find out that some authors call such spaces strongly S-closed spaces.

This basically answers my first question. But I am still keeping the question in my post - it is still possible that this class of spaces was studied under a different name or someone will be able to contribute some other interesting facts about such spaces.

Now we can define some similar topological properties:

Question 2. What about the spaces with this property: For any countable system of non-empty open sets $\{U_n; n=1,2,3,\dots\}$ such that $U_1\supseteq U_2\supseteq U_3\supseteq \dots$, the intersection $\bigcap U_n\ne\emptyset$. Let us call such spaces c.o.i. spaces. (Where "c.o.i." stands for countable open intersection).

Question 3. What is name/characterization of topological spaces with this property: $X$ has a base $\mathcal B$ such that for any countable system of sets $U_n\in\mathcal B$ such that $U_1\supseteq U_2\supseteq U_3\supseteq \dots$, the intersection $\bigcap U_n\ne\emptyset$. Let us call such spaces c.b.i. spaces. (Where "c.b.i." stands for countable basic intersection).

Question 4. What is name/characterization of topological spaces with this property: $X$ has a pseudobase ($\pi$-base) $\mathcal P$ such that for any countable system of sets $U_n\in\mathcal P$ such that $U_1\supseteq U_2\supseteq U_3\supseteq \dots$, the intersection $\bigcap U_n\ne\emptyset$. Let us call such spaces c.p.i. spaces. (Where "c.p.i." stands for countable pseudobasic intersection).

All these classes seem to be somehow related to Choquet game and Baire spaces.

It is clear that we have implications: u.i. space $\Rightarrow$ c.o.i. space $\Rightarrow$ c.b.i. space $\Rightarrow$ c.p.i space.

It can be shown that c.p.i. spaces are $\alpha$-favorable. (And it is known that every $\alpha$-favorable space is a Baire space.)

If we have a c.p.i. space, a tactic (stationary strategy) for the second player in Choquet game4 is easily obtained using the pseudobase with the c.p.i. property. If the first player plays some non-empty open sets $V_i$, the second player can choose a non-empty sets $U_i\subseteq V_i$ such that $U_i\in\mathcal P$. The property of the pseudobase $\mathcal P$ formulated in the definition of c.p.i. space implies that second player always wins, if he plays in this way.

Similarly, we can show that in a c.b.i. space, the second player has a winning strategy in strong Choquet game.

Several Baire extensions, which are studied in [HM, Section IV.2], are c.b.i. spaces.

1Open filters are defined as filters in the lattice of all open sets. I.e., the definition is almost the same as definition of filters, with the exception that we are working only with open sets. Open ultrafilters are defined as maximal open filters. Using Zorn Lemma we can show that any system of open sets, which has finite intersection property, is contained in an open ultrafilter.

2for example [M,Theorem 1], [HM,Proposition 4.13]

3 A pseudobase or a $\pi$-base of a topological space $X$ is a system $\mathcal P$ of non-empty open subsets of $X$ such that every open non-empty subset $U$ of $X$ contains some $V\in\mathcal P$.

4I am using the name strong Choquet game and Choquet game, similarly as in [K, Chapter 8]. Some basic fact about $\alpha$-favorable spaces are also mentioned in the article The Banach-Mazur Game at Dan Ma's topology blog, although the name "Choquet game" is not used there.

[D] J. Dontchev. Contra-continuous functions and strongly S-closed spaces. Internat. J. Math. Math. Sci., 19(2):303-310, 1996. DOI: 10.1155/S0161171296000427
[HM] R.C. Haworth and R.A. McCoy. Baire spaces. PWN, Warszawa, 1977. Dissertationes Mathematicae CXLI.
[K] A. S. Kechris. Classical descriptive set theory. Springer-Verlag, Berlin, 1995. Graduate Texts in Mathematics 156.
[M] R. A. McCoy. A Baire space extension. Proc. Amer. Math. Soc., 33(1):199-202, 1972. DOI: 10.1090/S0002-9939-1972-0293569-4, jstor