Editing Baire category theorem (section)

==Statement==
A [[Baire space]] is a topological space <math>X</math> in which every [[countable]] intersection of [[open (topology)|open]] [[dense set]]s is dense in <math>X.</math>  See the corresponding article for a list of equivalent characterizations, as some are more useful than others depending on the application.

* ('''BCT1''') Every [[complete metric space|complete]] [[pseudometric space]] is a Baire space.{{sfn|Kelley|1975|loc=theorem 34, p. 200}}{{sfn|Narici|Beckenstein|2011|loc=Theorem 11.7.2, p. 393}}{{sfn|Schechter|1996|loc=Theorem 20.16, p. 537}}  In particular, every [[completely metrizable]] topological space is a Baire space.{{sfn|Willard|2004|loc=Corollary 25.4}}
* ('''BCT2''') Every [[locally compact regular]] space is a Baire space.{{sfn|Kelley|1975|loc=theorem 34, p. 200}}{{sfn|Schechter|1996|loc=Theorem 20.18, p. 538}}  In particular, every [[locally compact]] [[Hausdorff space]] is a Baire space.{{sfn|Narici|Beckenstein|2011|loc=Theorem 11.7.3, p. 394}}{{sfn|Willard|2004|loc=Corollary 25.4}}

Neither of these statements directly implies the other, since there are complete metric spaces that are not locally compact (the [[irrational number]]s with the metric defined below; also, any [[Banach space]] of [[dimension of a vector space|infinite dimension]]), and there are locally compact Hausdorff spaces that are not [[Metrizable space|metrizable]] (for instance, any uncountable product of non-trivial compact Hausdorff spaces; also, several function spaces used in functional analysis; the uncountable [[Fort space]]).
See [[Counterexamples in Topology|Steen and Seebach]] in the references below.