Editing Baire category theorem (section)

==Proof==

('''BCT1''') The following is a standard proof that a complete pseudometric space <math>X</math> is a Baire space.{{sfn|Schechter|1996|loc=Theorem 20.16, p. 537}}

Let <math>U_1, U_2, \ldots</math> be a countable collection of open dense subsets. We want to show that the intersection <math>U_1 \cap U_2 \cap \ldots</math> is dense.
A subset is dense if and only if every nonempty open subset intersects it. Thus to show that the intersection is dense, it suffices to show that any nonempty open subset <math>W</math> of <math>X</math> has some point <math>x</math> in common with all of the <math>U_n</math>.
Because <math>U_1</math> is dense, <math>W</math> intersects <math>U_1;</math> consequently, there exists a point <math>x_1</math> and a number <math>0 < r_1 < 1</math> such that:
<math display=block>\overline{B}\left(x_1, r_1\right) \subseteq W \cap U_1</math>
where <math>B(x, r)</math> and <math>\overline{B}(x, r)</math> denote an open and closed ball, respectively, centered at <math>x</math> with radius <math>r.</math>
Since each <math>U_n</math> is dense, this construction can be continued recursively to find a pair of sequences <math>x_n</math> and <math>0 < r_n < \tfrac{1}{n}</math> such that:
<math display=block>\overline{B}\left(x_n, r_n\right) \subseteq B\left(x_{n-1}, r_{n-1}\right) \cap U_n.</math>

(This step relies on the axiom of choice and the fact that a finite intersection of open sets is open and hence an open ball can be found inside it centered at <math>x_n</math>.)
The sequence <math>\left(x_n\right)</math> is [[Cauchy sequence|Cauchy]] because <math>x_n \in B\left(x_m, r_m\right)</math> whenever <math>n > m,</math> and hence <math>\left(x_n\right)</math> converges to some limit <math>x</math> by completeness.
If <math>n</math> is a positive integer then <math>x \in \overline{B}\left(x_n, r_n\right)</math> (because this set is closed). 
Thus <math>x \in W</math> and <math>x \in U_n</math> for all <math>n.</math> <math>\blacksquare</math>

There is an alternative proof using [[Choquet's game]].<ref>{{cite web|last=Baker|first=Matt|title=Real Numbers and Infinite Games, Part II: The Choquet game and the Baire Category Theorem|url=https://mattbaker.blog/2014/07/07/real-numbers-and-infinite-games-part-ii/#more-733|date=July 7, 2014}}</ref>

('''BCT2''') The proof that a [[locally compact regular]] space <math>X</math> is a Baire space is similar.{{sfn|Schechter|1996|loc=Theorem 20.18, p. 538}}  It uses the facts that (1) in such a space every point has a [[local base]] of [[closed (topology)|closed]] [[compact (topology)|compact]] neighborhoods; and (2) in a compact space any collection of closed sets with the [[finite intersection property]] has nonempty intersection.  The result for [[locally compact Hausdorff]] spaces is a special case, as such spaces are regular.