We present a useful characterization of Baire spaces. A Baire space is a topological space in which the conclusion of Baire category theorem holds, that is, for each countable family of open and dense subsets of , the intersection is dense in . This definition is equivalent to the statement that every non-empty open subset of is of second category in . 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  and ). We prove the following theorem.
A collection of subsets of a space is said to be point-finite if every point in the space belongs to at most finitely many members of . A collection of subsets of is said to be locally finite at the point if there is an open set such that and meets at most finitely many members of . The collection is said to be locally finite in the space if it is locally finite at every . For any terms and concepts not explicitly defined here, refer to  (Engelking) or ) (Willard).
Let be a space. The following conditions are equivalent.
- is a Baire space.
- For any point-finite open cover of , the set is a dense set in .
- For any countable point-finite open cover of , the set is a dense set in .
Let be a point-finite open cover of . Let be a non-empty open subset of . We wish to show that where is the set defined in condition 2. For each , define
Note that . Since is a Baire space, must be of second category in . None of the sets can be a nowhere dense set. Thus for some , has non-empty interior. Choose some non-empty open set such that .
Pick . Since , let be the members of that contain . Let . Note that . Observe that is a non-empty open set that meets exactly members of . Therefore is locally finite at points of , leading to the conclusion that and .
The direction is immediate.
Suppose condition 3 holds. We claim that is a Baire space. Suppose not. Let be a non-empty open subset of such that where each is nowhere dense in . Let be defined as the following:
where . Clearly, is a point-finite open cover of . By condition 3, is dense in ( is defined in condition 3). In particular, . Choose . Since is locally finite at , we can choose some open set such that and such that meets only finitely many , say only up to (so for all ).
On the other hand, all sets are nowhere dense. So we can choose some open set such that misses the nowhere dense set . In particular, this means that , contradicting that for all . So must be a Baire space if condition 3 holds.
- Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
- Fletcher, P., Lindgren, W. F., A note on spaces of second category, Arch, Math., 24, 186-187, 1973.
- McCoy, R. A., A Baire space extension, Proc. Amer. Math. Soc., 33, 199-202, 1972.
- Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.