Are the two formulations of Baire category theorem equivalent for arbitrary metric spaces

baire-categoryfunctional-analysisgeneral-topology

As we all know, Baire Category theorem has two equivalent forms

  1. $X$ is a complete metric space, then the countable intersection of dense open sets is nonempty.
  2. $X$ is a complete metric space, $X$ is a second category set.

Two forms are equivalent if $X$ is complete. If $X$ is a general metric space, are they still equivalent?

Best Answer

The general equivalence is just (for any topological space, not just metrisable ones)

$X$ is a Baire space iff $X$ is second category in itself.

E.g. see Willard (25.2). Just note that a set of nowhere dense iff its complement contains a dense open set. A Baire space is a space where any countable intersection of dense open subsets is dense.

Then there is a separate theorem that a complete metric space is a Baire space, so that the above general equivalence applies to them as well.

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