This post deals with a classical problem in functional analysis: The Baire space. I am not going to reproduce what we can learn and read from wikipedia; however, to make the post self-contained, following is what the Baire space is.

Loosely speaking, a Baire space is a topological space in which any one of the following three equivalent conditions is satisfied:

- Whenever the union of countably many closed subsets of has an interior point, then one of the closed subsets must have an interior point, i.e. if
then for some . Here by we mean a closed subset in .

- The union of every countable collection of closed sets with empty interior has empty interior, that is to say, i.e if for all , then
- Every intersection of countably many dense open sets is dense, i.e.
provided for every . Here by we mean an open subset in .

What I am going to do is to show that every open subset of a Baire space is itself a Baire space, of course, under the subspace topology inherited from . Hence, at the very beginning, we assume throughout this topic that is a Baire space; hence admits all three equivalent conditions above.