From 28407333ffceca9b99fae721c30e8ae146a863da Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Wed, 14 Feb 2024 07:24:38 +0100 Subject: Update --- pages/general-topology/baire-spaces.md.txt | 73 ------------------------------ 1 file changed, 73 deletions(-) delete mode 100644 pages/general-topology/baire-spaces.md.txt (limited to 'pages/general-topology/baire-spaces.md.txt') diff --git a/pages/general-topology/baire-spaces.md.txt b/pages/general-topology/baire-spaces.md.txt deleted file mode 100644 index eabe792..0000000 --- a/pages/general-topology/baire-spaces.md.txt +++ /dev/null @@ -1,73 +0,0 @@ ---- -title: Baire Spaces -parent: General Topology -nav_order: 1 -description: > - A Baire space is a topological space with the property that the intersection - of countably many dense open subsets is still dense. One version of the Baire - Category Theorem states that complete metric spaces are Baire spaces. We give - a self-contained proof of Baire's Category Theorem by contradiction. -# spellchecker:words ---- - -# - - -A topological space is said to be a *Baire space*, -if any of the following equivalent conditions holds: -> -- The intersection of countably many dense open subsets is still dense. -- The union of countably many closed subsets with empty interior has empty interior. - - -Note that -a set is dense in a topological space -if and only if -its complement has empty interior. - -Any sufficient condition -for a topological space to be a Baire space -constitutes a *Baire Category Theorem*, -of which there are several. -Here we give one -that is commonly used in functional analysis. - - -Baire Category Theorem - -> -Complete metric spaces are Baire spaces. - -**Proof:** -Let C-C-C be a metric space -with complete metric D-D-D. -Suppose that F-F-F is not a Baire space. -Then there is a countable collection G-G-G of dense open subsets of B-B-B -such that the intersection C-C-C is not dense in D-D-D. - -In a metric space, any nonempty open set contains an open ball. -It is also true, that any nonempty open set contains a closed ball, -since F-F-F if G-G-G. - -We construct a sequence B-B-B of open balls C-C-C satisfying - -V-V-V -as follows: By hypothesis, -the interior of D-D-D is not empty (otherwise F-F-F would be dense in G-G-G), -so we may choose an open ball B-B-B with C-C-C -such that D-D-D. -Given F-F-F, -the set G-G-G is nonempty, because B-B-B is dense in C-C-C, -and it is open, because D-D-D and F-F-F are open. -This allows us to choose an open ball G-G-G as desired. - -Note that by construction B-B-B for C-C-C, -thus D-D-D. -Therefore, the sequence F-F-F is Cauchy -and has a limit point G-G-G by completeness. -In the limit B-B-B, we obtain C-C-C (strictness is lost), -hence D-D-D for all F-F-F. -This shows that G-G-G for all B-B-B, that is C-C-C. -On the other hand, D-D-D, -in contradiction to the preceding statement. - -- cgit v1.2.3-54-g00ecf