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.
Read more about this topic: Baire Space
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