A Characterization of Baire Spaces

We present a useful characterization of Baire spaces. A Baire space is a topological space X in which the conclusion of Baire category theorem holds, that is, for each countable family \left\{U_1,U_2,U_3,\cdots \right\} of open and dense subsets of X, the intersection \bigcap \limits_{n=1}^\infty U_n is dense in X. This definition is equivalent to the statement that every non-empty open subset of X is of second category in X. An elementary discussion of Baire spaces is found in this blog post. Baire spaces can also be characterized in terms of the Banach-Mazur game (see Theorem 1 in this post). We add one more characterization in terms of point-finite open cover and locally finite open family (from [2] and [3]). We prove the following theorem.

A collection \mathcal{S} of subsets of a space X is said to be point-finite if every point in the space X belongs to at most finitely many members of \mathcal{S}. A collection \mathcal{S} of subsets of X is said to be locally finite at the point x \in X if there is an open set V \subset X such that x \in V and V meets at most finitely many members of \mathcal{S}. The collection \mathcal{S} is said to be locally finite in the space X if it is locally finite at every x \in X. For any terms and concepts not explicitly defined here, refer to [1] (Engelking) or [4]) (Willard).

Theorem
Let X be a space. The following conditions are equivalent.

  1. X is a Baire space.
  2. For any point-finite open cover \mathcal{U} of X, the set D=\left\{x \in X: \mathcal{U} \text{ is locally finite at } x \right\} is a dense set in X.
  3. For any countable point-finite open cover \mathcal{U} of X, the set D=\left\{x \in X: \mathcal{U} \text{ is locally finite at } x \right\} is a dense set in X.

Proof
1 \Rightarrow 2
Let \mathcal{U} be a point-finite open cover of X. Let O be a non-empty open subset of X. We wish to show that O \cap D \ne \varnothing where D is the set defined in condition 2. For each n, define

\displaystyle . \ \ \ \ \ F_n=\left\{x \in O: x \text{ belongs to exactly n members of } \mathcal{U} \right\}.

Note that O=\bigcup \limits_{n=1}^\infty F_n. Since X is a Baire space, O must be of second category in X. None of the sets F_n can be a nowhere dense set. Thus for some n, F_n has non-empty interior. Choose some non-empty open set W such that W \subset F_n.

Pick y \in W. Since y \in F_n, let U_1,U_2,\cdots,U_n be the n members of \mathcal{U} that contain y. Let V=W \cap \bigcap \limits_{j=1}^n U_n. Note that V \subset W \subset F_n \subset O. Observe that V is a non-empty open set that meets exactly n members of \mathcal{U}. Therefore \mathcal{U} is locally finite at points of V, leading to the conclusion that V \subset D and O \cap D \ne \varnothing.

The direction 2 \Rightarrow 3 is immediate.

3 \Rightarrow 1
Suppose condition 3 holds. We claim that X is a Baire space. Suppose not. Let U be a non-empty open subset of X such that U=\bigcup \limits_{n=1}^\infty K_n where each K_n is nowhere dense in X. Let \mathcal{U} be defined as the following:

\displaystyle . \ \ \ \ \ \mathcal{U}=\left\{X \right\} \cup \left\{U_n: n=1,2,3,\cdots\right\},

where U_n=U - (\overline{K_1} \cup \cdots \cup \overline{K_n}). Clearly, \mathcal{U} is a point-finite open cover of X. By condition 3, D is dense in X (D is defined in condition 3). In particular, U \cap D \ne \varnothing. Choose y \in U \cap D. Since \mathcal{U} is locally finite at y, we can choose some open set V \subset U such that y \in V and such that V meets only finitely many U_j, say only up to U_1,\cdots, U_m (so V \cap U_j = \varnothing for all j > m).

On the other hand, all sets K_j are nowhere dense. So we can choose some open set V_0 \subset V such that V_0 misses the nowhere dense set \overline{K_1} \cup \cdots \cup \overline{K_m} \cup \overline{K_{m+1}}. In particular, this means that V_0 \cap U_{m+1} \ne \varnothing, contradicting that V \cap U_j = \varnothing for all j > m. So X must be a Baire space if condition 3 holds. \blacksquare

Reference

  1. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
  2. Fletcher, P., Lindgren, W. F., A note on spaces of second category, Arch, Math., 24, 186-187, 1973.
  3. McCoy, R. A., A Baire space extension, Proc. Amer. Math. Soc., 33, 199-202, 1972.
  4. Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.
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