**EDIT:** I have reposted this question on MathOverflow. (The version posted there is more concise, with some details omitted. I have also added a question about pseudobases with similar property.)

MO copy: Intersections of open sets and $\alpha$-favorable spaces

I was curious a little about the classes of topological spaces described below. For each of them I want to ask whether they have been studied and whether they have some interesting properties. (Although I do not think that they are going to be particularly useful, since all these properties seem to be more natural if the intersections of open sets in the definitions below are replaced by intersections of their closures.) For the purpose of this post I have made up some names for these classes of spaces.

I know that my question is a little vague - I am basically asking whether these classes were studied somewhere, what are they called (if they have a name) and whether they have some interesting properties; although it is rather subjective what can be considered interesting.

Below I will try to write down what I was able to find out by myself and I will also try to give some motivation for the question.

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 spacesu.i.-spaces. (Where "u.i." stands for ultrafilter intersection).^{1}

Question 2. What is name/characterization of spaces such that $\bigcap \mathcal F\ne\emptyset$ for every open filter $\mathcal F$. Let us call such spacesf.i.-spaces. (Where "f.i." stands for filter intersection).

I will try to show below that the conditions f.i. and u.i. are in fact equivalent. And I will also give a reformulation, which is more similar to the conditions in the following two questions. (I hope I have not made a mistake in the proof below.) But the above formulation is closer to the result from [HM], which motivated me to ask about these spaces, so I have kept this formulation. I was able to find some papers, where this class of spaces is mentioned.

Question 3. 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 spacesc.o.i. spaces. (Where "c.o.i." stands for countable open intersection).

Question 4. 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 spacesc.b.i. spaces. (Where "c.b.i." stands for countable basic intersection).

All these classes seem to be related to Baire spaces somehow. The reason why I wondered what can be said about these classes of spaces was the result that u.i. spaces are Baire spaces, which I mention below.

**Some basic observations**

**Reformulation using closed covers.** It is clear that in the definition of c.o.i. space we can replace decreasing countable family of non-empty open sets by a countable family of open sets with finite intersection property. By a standard argument we see that this is equivalent to: Every countable cover of $X$ by closed sets has a finite subcover.

**Equivalence of f.i. and u.i.** Similar condition for all closed covers (not only countable ones) was discussed a little in one of the answers to this question: Terminologies related to "compact?" It seems that some authors call such spaces *strongly S-closed spaces*. In [D, Theorem 3.3] it is shown that an equivalent characterization is that there exists a finite dense subset in this space.

Again, using the standard argument (taking complements, similarly as for compact spaces) we get that this condition is equivalent to: Every system of open sets which has finite intersection property has non-empty intersection.

I'll try to show that the following three conditions are equivalent:

- (a) Every system of open subsets of $X$ which has finite intersection property has a non-empty intersection.
- (b) Every open filter on $X$ has a non-empty intersection.
- (c) Every open ultrafilter on $X$ has a non-empty intersection.

By definition, an open filter has f.i.p. Hence (a) $\Rightarrow$ (b). It is clear that (b) $\Rightarrow$ (c).

$\boxed{\text{(c)}\Rightarrow\text{(a)}}$ If $\mathscr S$ is a system of open sets with f.i.p. then (by Zorn lemma) there exists an open ultrafilter $\mathscr F\supseteq\mathscr S$. Since $\mathscr F$ has non-empty intersection, so does $\mathscr S$.

So we get that the notions of u.i. spaces and f.i. spaces are equivalent to the notion of strongly S-closed spaces. (Which basically answers my first two questions. But I am still keeping these two questions in my post - it is still possible that I have made a mistake in my proof, and also 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.)

As pointed out in a (now deleted) comment by Niels Diepeveen, $T_1$-spaces have this property if and only if they are finite. (Closed cover by singletons has to have a finite subcover.)

**c.o.i. and c.b.i. are not equivalent conditions.** As an example we can take an infinite discrete space. The base consisting of all singletons has the property described above.

So far we see that f.i $\Leftrightarrow$ u.i. $\Rightarrow$ c.o.i $\Rightarrow$ c.b.i.

**Background, motivation, related notions**

**u.i.-spaces are Baire spaces.** These classes of spaces caught my attention after I have seen the following result^{2}, 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.)

**c.b.i. spaces are $\alpha$-favorable.** The classes of c.o.i. and c.b.i. are also related to Baire spaces. It can be shown that in c.b.i. the second player has a winning tactic for Choquet game. Choquet game can be used to characterize Baire spaces, they are precisely the spaces where the first player does not have a winning strategy^{3} (or, equivalently, the first player does not have a winning tactic^{4}).

It is relatively easy to see that every c.b.i. space is *$\alpha$-favorable.* (And every $\alpha$-favorable space is a Baire space.) The tactic (stationary strategy) for the second player in Choquet game is easily obtained using the base with the c.b.i. property. If the first player plays some non-empty open sets $V_i$, the second player can choose a non-empty basic sets $U_i\subseteq V_i$. The property of the base $\mathcal B$ formulated in the definition of c.b.i. space implies that second player always wins, if he plays in this way.

**Some other related properties.** Some similar properties are studied if we require non-empty intersection of closures of the open sets from a filter/countable decreasing system.

The spaces such that $\bigcap_{U\in\mathscr F} \overline U\ne\emptyset$ for each open filter $\mathscr F$ are called *generalized absolutely closed spaces* or *almost compact spaces*. If we add $T_2$, we get precisely H-minimal spaces.

A space which is quasi-regular has a base $\mathcal B$ such that $\bigcap_{U\in\mathcal U} \overline U\ne\emptyset$ for any countable system $\mathcal U\subseteq\mathcal B$ with finite intersection property is called *countably base compact*. (Although this property does not seem to appear too often, more references can be found for base compact spaces, where the countability condition is omitted.)

**The extension $sX$.** One of the spaces studied in [HM, Section IV.2] is the space called $sX$, which is constructed from $X$ as follows: We add to $X$ as new points the set $F$ of all open ultrafilters, that are generated by countable systems of open sets $\{U_n; n=1,2,\dots\}$ such that $U_1\supseteq U_2 \supseteq \dots$ and $\bigcap\limits_{i=1}^\infty U_i=\emptyset$. We take the topology given by the base consisting of sets $U^*=U\cup\{\mathfrak F\in F; U\in\mathfrak F\}$.

It is shown that $sX$ is a Baire space which contains $X$ as a dense subspace and also some other results about this space are mentioned there.

Clearly a space $X$ is an c.o.i. space if and only if $sX=X$. (But this is more of a reformulation than a characterization.)

The reasons I am mentioning $sX$ is that it is an example of a c.b.i. space. (And so are some other extensions mentioned in that chapter.)

Suppose we have basic sets $U_1^*\supseteq U_2^* \supseteq U_3 \supseteq \dots$. We can assume that $U_1\supseteq U_2 \supseteq U_3 \supseteq \dots$. (If not, we simply take $U_n' = \bigcap_{k=1}^n U_k$. This does not change the decreasing system of basic sets, since $(U\cap V)^*=U^*\cap V^*$ and the same is true for any finite intersection.) Then there are two possibilities. If $\bigcap\limits_{i=1}^\infty U_i\ne\emptyset$, then also the intersection $\bigcap\limits_{i=1}^\infty U_i^*$ is non-empty. On the other hand, if $\bigcap\limits_{i=1}^\infty U_i=\emptyset$, then the filter $\mathfrak F$ generated by $\{U_i; i=1,2,\dots\}$ is a point of $sX$ which belongs to $\bigcap\limits_{i=1}^\infty U_i^*$.

[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

^{1}Open 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.

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

^{3}See, for example, [K,Theorem 8.11] or The Banach-Mazur Game at Dan Ma's topology blog.

^{4}~~I thought that the proof of [K, Theorem 8.11] in fact yields the same result for stationary strategy (instead of strategy). Now I have noticed that this is not true. As I do not know whether the equivalence of these two conditions is true, I have removed the part about winning tactic.~~ The equivalence of these two conditions is shown in Corollary 1' of Fred Galvin, Ratislav Telgársky: Stationary strategies in topological games, Topology and its Applications, Volume 22, Issue 1, February 1986, Pages 51–69, DOI: 10.1016/0166-8641(86)90077-5.