The Baire category theorem, also known as Baire's theorem and the category theorem, is a result in analysis and set theory which roughly states that in certain spaces, the intersection of any countable collection of "large" sets remains "large." The appearance of "category" in the name refers to the interplay of the theorem with the notions of sets of first and second category.
Precisely stated, the theorem says that if a space 
is either a complete
metric space or a locally compact T2-space,
then the intersection of every countable
collection of dense open subsets of 
is necessarily dense in 
.
The above-mentioned interplay with first category and second category sets
can be summarized by a single corollary, namely that nonempty spaces 
that are either complete
metric spaces or locally compact Hausdorff
spaces are of second category in themselves. To
see that this follows from the above-stated theorem, let 
be either a complete metric
space or a locally compact Hausdorff space
and note that if 
is a countable collection of nowhere
dense subsets of 
and if 
denotes the complement
in 
of the closure 
of 
, then each set 
is necessarily dense in 
. Because of the theorem, it follows that the intersection
of all the sets 
must be nonempty (and indeed
must be dense in 
), thereby proving that 
cannot be written as the union of
the sets 
. In particular, such nonempty spaces 
cannot be written as a countable union of subsets that are nowhere
dense in 
.
Such spaces are therefore of second
category relative to themselves.
