Baire Space - Baire Category Theorem

Baire Category Theorem

The Baire category theorem gives sufficient conditions for a topological space to be a Baire space. It is an important tool in topology and functional analysis.

• (BCT1) Every complete metric space is a Baire space. More generally, every topological space which is homeomorphic to an open subset of a complete pseudometric space is a Baire space. In particular, every topologically complete space is a Baire space.
• (BCT2) Every locally compact Hausdorff space is a Baire space.

BCT1 shows that each of the following is a Baire space:

• The space R of real numbers
• The space of irrational numbers
• The Cantor set
• Indeed, every Polish space

BCT2 shows that every manifold is a Baire space, even if it is not paracompact, and hence not metrizable. For example, the long line is of second category.