Baire Category Theorem

Theorem

The consists of two statements, each of which is independent of the other:

Let $M = \struct {A, d}$ be a complete metric space.

Then $M = \struct {A, d}$ is also a Baire space.

Let $T = \struct {S, \tau}$ be a Hausdorff space.

Let $T$ be locally compact.

Then $T = \struct {S, \tau}$ is also a Baire space.

This entry was named for René-Louis Baire.

The has important uses in functional analysis.

One of its uses is implying that an algebraic or Hamel basis of an infinite dimensional Banach space must be uncountable.

Hence the need for Schauder bases.

2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Baire Category Theorem