Something about the Baire Space

Question

I do not know why we put forward the "Baire Space"? What is the difference between the Baire Space and Metric Space? Can you give me some examples? Thank you very much!

Answer 1

The concept of a Baire space is the topological generalization of the concept of a complete metric space. Specifically, complete metric spaces have the property that a countable intersection of open dense sets is still dense. (This statement is called the Baire category theorem, or at least a version of it.) Baire spaces are those topological spaces that also have this property. The most basic type of Baire space other than a complete metric space is a locally compact Hausdorff space.

For examples of why we would care, I'd suggest you look up applications of the Baire category theorem. I could state some interesting results of this type but it would probably not be obvious to you where the Baire category theorem enters into their proofs.