Baire Category Theorem and the Finite Intersection Property

A Baire space is a topological space in which the intersection of any countable family of open and dense sets is dense (equivalently every non-empty open subset is of second category). One version of the Baire category theorem states that every complete metric space is a Baire space. Another common version states that every compact Hausdorff space is a Baire space. Another version states that every locally compact Hausdorff space is a Baire space. The commonality among these versions is the finite intersection property (whenever a collection of a certain type of sets satisfies the property that any finite subcollection has non-empty intersection, the whole collection has non-empty intersection). For each of these classes of spaces, in addition to countably compact spaces and pseudocompact spaces, Baire category theorem is derived from having one specific form of the finite intersection property. In this post, we explore this relationship.

In each of the following theorem pairs, the B Theorem follows from the A theorem. The A theorem is a form of the finite intersection property and the B theorem is a version of Baire category theorem.

Another interesting observation is that the finite intersection properties discussed here can give a stronger property than being a Baire space. This stronger property is defined by the Banach-Mazur game.

__________________________________________________________________________

Theorem 1A
Let (X, \rho) be a metric space. Then the following conditions are equivalent.

  1. (X, \rho) is a complete metric space.
  2. For each decreasing sequence C_1 \supset C_2 \supset C_3 \supset \cdots of non-empty closed subsets of X such that the diameters of the sets C_n converge to zero, we have \bigcap \limits_{n=1}^\infty C_n \ne \varnothing.

Theorem 1B
Every complete metric space is a Baire space.

__________________________________________________________________________

Theorem 2A
Let X be a Hausdorff space. Then the following conditions are equivalent.

  1. X is a compact space.
  2. For every family \mathcal{F} consisting of non-empty closed subsets of X, if \mathcal{F} has the finite intersection property, then \mathcal{F} has non-empty intersection.

Theorem 2B

  • Every compact Hausdorff space is a Baire space.
  • Every locally compact Hausdorff space is a Baire space.

__________________________________________________________________________

Theorem 3A
Let X be a Hausdorff space. Then the following conditions are equivalent.

  1. X is a countably compact space.
  2. For every countable family \mathcal{F} consisting of non-empty closed subsets of X, if \mathcal{F} has the finite intersection property, then \mathcal{F} has non-empty intersection.
  3. For each decreasing sequence C_1 \supset C_2 \supset C_3 \supset \cdots of non-empty closed subsets of X, we have \bigcap \limits_{n=1}^\infty C_n \ne \varnothing.

Theorem 3B
Every countably compact Hausdorff space is a Baire space.

__________________________________________________________________________

Theorem 4A
Let X be a regular space. The following conditions are equivalent:

  1. The space X is pseudocompact.
  2. If \mathcal{O}=\left\{O_1,O_2,O_3,\cdots \right\} is a family of non-empty open subsets of X such that O_n \supset O_{n+1} for each n, then \bigcap \limits_{n=1}^\infty \overline{O_n} \ne \varnothing.
  3. If \mathcal{V}=\left\{V_1,V_2,V_3,\cdots \right\} is a family of non-empty open subsets of X such that \mathcal{V} has the finite intersection property, then \bigcap \limits_{n=1}^\infty \overline{V_n} \ne \varnothing.

Theorem 4B
Every regular pseudocompact space is a Baire space.

__________________________________________________________________________

Remark
Theorem 1A (the Cantor Theorem) can be found in Engelking (page 269 in [1]). Theorem 2A and Theorem 3A can also be found in Engelking (they are also proved in this post). Theorem 4B is also found in Engelking (Theorem 3.10.23 in page 207 of [1]) and is proved this post.

We would like to explicitly point out that between Thoerem 1A and Theorem 2A, none of the two theorems implies the other. For example, even though both complete metric spaces and compact Hausdorff spaces are Baire spaces, complete metric spaces are not necessarily compact and there are compact spaces that are not even metrizable. However, the finite intersection property of Theorem 2A implies that of Theorem 3A, which in turn implies the finite intersection property of Theorem 4A.
__________________________________________________________________________

Baire Category Theorem

The proofs of all four B theorems are amazingly similar. It is a matter of exploiting the fact that whenever a decreasing sequence of open sets satisfying the condition that each closure is a subset of the previous open set (and satisfying some other condition), the sequence of open sets has non-empty intersection. For example, for complete metric space, make sure that the closures of the open sets have diameters going to zero. For any reader who is new to this material, it will be very instructive to walk through the arguments of these Baire category theorems. The proof of Theorem 1A can be found this post. We prove Theorem 4B.

Recall that X is a Baire space if \left\{U_1,U_2,U_3,\cdots \right\} is a countable family of open and dense sets in X, \bigcap \limits_{i=1}^\infty U_i is dense in X, or equivalently every non-empty open subset of X is of second category in X. For more background about the concepts of Baire space and category (see [1] or this post).

Proof of Theorem 4B
Let X be a regular pseudocompact space. Let \left\{U_1,U_2,U_3,\cdots \right\} be a countable family of open and dense sets in X. Let O be a non-empty open subset of X. We show that O has to contain points of \bigcap \limits_{n=1}^\infty U_n. We let O_1=O \cap U_1. We find open O_2 such that O_2 \subset U_2 and \overline{O_2} \subset O_1 (using regularity). Continue this inductive process, we have for each n, an open O_n such that O_n \subset U_n and \overline{O_n} \subset O_{n-1}. Then we have a decreasing sequence of open sets O_n as in condition 2 of Theorem 4A. Then we have \bigcap \limits_{n=1}^\infty \overline{O_n} \ne \varnothing. Since \overline{O_{n+1}} \subset O_n for each n, we also have \bigcap \limits_{n=1}^\infty O_n \ne \varnothing. It is clear that \bigcap \limits_{n=1}^\infty O_n \subset \bigcap \limits_{n=1}^\infty U_n. \blacksquare

__________________________________________________________________________

Banach-Mazur Game

In proving the above versions of Baire category theorem, we can exploit the appropriate version of the finite intersection property – the situation that any nested decreasing sequence of open sets (under some specified conditions) has non-empty intersection. In fact, the finite intersection property offers more than just Baire category theorem; it can endow the space in question a type of completeness property stronger than Baire space. This completeness property is defined using the Banach-Mazur game.

The Banach-Mazur game is a two-person game played on a topological space. Let X be a space. There are two players, \alpha and \beta. They take turn choosing nested decreasing nonempty open subsets of X as follows. The player \beta goes first by choosing a nonempty open subset U_0 of X. The player \alpha then chooses a nonempty open subset V_0 \subset U_0. At the nth play where n \ge 1, \beta chooses an open set U_n \subset V_{n-1} and \alpha chooses an open set V_n \subset U_n. The player \alpha wins if \bigcap \limits_{n=0}^\infty V_n \ne \varnothing. Otherwise the player \beta wins. For more detailed discussion of the game, see this post.

One interesting point that we like to make about the finite intersection property ranging from Theorem 1A to Theorem 4A is that the player \alpha can always win the Banach-Mazur game as long as he/she plays the game according to each specific version of the finite intersection. For example, playing the game in a complete metric space, player \alpha always wins as long as he/she makes the diameters of the closures of the open sets going to zero. In a regular pseudocompact space, player \alpha can always win by making the closure of each of his/her open sets a subset of the previous move of other player.

A topological space in which the player \alpha has a winning strategy is said to be a weakly \alpha-favorable space. Thus complete metric spaces, compact Hausdorff spaces, locally compact Hausdorff spaces, countably compact Hausdorff spaces, regular pseudocompact spaces are all weakly \alpha-favorable.

There is characterization of Baire spaces in terms of the Banach-Mazur game. A space X is a Baire space if and only if the player \beta has no winning strategy in the Banach-Mazur game played on the space X (see theorem 1 in this post). If the player \alpha can always win, then player \beta can never win. In terms of game terminology, if player \alpha has a winning strategy, then the other player (player \beta) has no winning strategy. Thus a space is weakly \alpha-favorable implies that it is a Baire space. But the implication is not reversible (see example in this post).

So all the spaces discussed from Theorem 1A to Theorem 4A are all weakly \alpha-favorable, a property stronger than Baire spaces. These observations are summarized in the following theorems.

Theorem 1C
Every complete metric space is a weakly \alpha-favorable space.

Theorem 2C

  • Every compact Hausdorff space is a weakly \alpha-favorable space.
  • Every locally compact Hausdorff space is a weakly \alpha-favorable space.

Theorem 3C
Every countably compact Hausdorff space is a weakly \alpha-favorable space.

Theorem 4C
Every regular pseudocompact space is a weakly \alpha-favorable space.

__________________________________________________________________________

Reference

  1. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
  2. Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.
Advertisements

One thought on “Baire Category Theorem and the Finite Intersection Property

  1. We think that Theorem 3B is wrong. In the frame of its proof “Hausdorff” should be replaced to “regular” (or, at least, to “quasiregular”).

    Oleg Okunev and Eugenii Reznichenko built a countably compact Hausdorff non Baire space $X$. Moreover, this space is $\omega$-bounded, that is each countable subset of $X$ has the compact closure.

    Their proof is based on the following idea. Let $(X,\tau)$ be a countably compact Hausdorff space and $X=\bigcup X_n$ be a union of a monotonically increasing sequence of countably compact subsets such that each $X_n$ has the empty interior. The they added to the topology on the set $X$ the sets of the form $X\setminus X_n$.

    They put $X_n=\bigcup\{Y_k:k\le n\}$, where $Y_k\subset [0,1]^{\omega_1}$ – this is the $\Sigma$-product of $\omega_1$ segments centered at the point having all the coordinates equal to $1/k$, provided $k>0$, and at the zero, provided $k=0$.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s