From 777f9d3fd8caf56e6bc6999a4b05379307d0733f Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Tue, 12 Sep 2023 07:36:33 +0200 Subject: Initial commit --- pages/general-topology/baire-spaces.md | 83 ++++++++++++++++++++++ pages/general-topology/baire-spaces.md.txt | 73 +++++++++++++++++++ pages/general-topology/compactness/basics.md | 43 +++++++++++ pages/general-topology/compactness/index.md | 9 +++ .../compactness/tychonoff-product-theorem.md | 19 +++++ pages/general-topology/index.md | 11 +++ pages/general-topology/jordan-curve-theorem.md | 18 +++++ 7 files changed, 256 insertions(+) create mode 100644 pages/general-topology/baire-spaces.md create mode 100644 pages/general-topology/baire-spaces.md.txt create mode 100644 pages/general-topology/compactness/basics.md create mode 100644 pages/general-topology/compactness/index.md create mode 100644 pages/general-topology/compactness/tychonoff-product-theorem.md create mode 100644 pages/general-topology/index.md create mode 100644 pages/general-topology/jordan-curve-theorem.md (limited to 'pages/general-topology') diff --git a/pages/general-topology/baire-spaces.md b/pages/general-topology/baire-spaces.md new file mode 100644 index 0000000..6bd7d9f --- /dev/null +++ b/pages/general-topology/baire-spaces.md @@ -0,0 +1,83 @@ +--- +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 +--- + +# {{ page.title }} + +{: .definition } +> A topological space is said to be a *Baire space*, +> if any of the following equivalent conditions holds: +> {: .mb-0 } +> +> - The intersection of countably many dense open subsets is still dense. +> - The union of countably many closed subsets with empty interior has empty interior. +> {: .mt-0 .mb-0 } + +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. + +{: .theorem-title } +> Baire Category Theorem #1 +> {: #baire-category-theorem } +> +> Every complete metric space is a Baire space. + +{% proof %} +Let $X$ be a metric space +with complete metric $d$. +Suppose that $X$ is not a Baire space. +Then there is a countable collection $\braces{U_n}$ of dense open subsets of $X$ +such that the intersection $U := \bigcap U_n$ is not dense in $X$. + +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 $\overline{B(y,\delta_1)} \subset B(y,\delta_2)$ if $\delta_1 < \delta_2$. + +We construct a sequence $(B_n)$ of open balls $B_n := B(x_n,\epsilon_n)$ satisfying + +$$ +\overline{B_{n+1}} \subset B_n \cap U_n \quad \epsilon_n < \tfrac{1}{n} \quad \forall n \in \NN, +$$ + +as follows: By hypothesis, +the interior of $X \setminus U$ is not empty (otherwise $U$ would be dense in $X$), +so we may choose an open ball $B_1$ with $\epsilon_1 < 1$ +such that $\overline{B_1} \subset X \setminus U$. +Given $B_n$, +the set $B_n \cap U_n$ is nonempty, because $U_n$ is dense in $X$, +and it is open, because $B_n$ and $U_n$ are open. +This allows us to choose an open ball $B_{n+1}$ as desired. + +Note that by construction $B_m \subset B_n$ for $m \ge n$, +thus $d(x_m,x_n) < \epsilon_n < \tfrac{1}{n}$. +Therefore, the sequence $(x_n)$ is Cauchy +and has a limit point $x$ by completeness. +In the limit $m \to \infty$, we obtain $d(x,x_n) \le \epsilon_n$ (strictness is lost), +hence $x \in \overline{B_n}$ for all $n$. +This shows that $x \in U_n$ for all $n$, that is $x \in U$. +On the other hand, $x \in \overline{B_1} \subset X \setminus U$, +in contradiction to the preceding statement. +{% endproof %} + +{: .theorem-title } +> Baire Category Theorem #2 +> {: #baire-category-theorem } +> +> Every compact Hausdorff space is a Baire space. diff --git a/pages/general-topology/baire-spaces.md.txt b/pages/general-topology/baire-spaces.md.txt new file mode 100644 index 0000000..eabe792 --- /dev/null +++ b/pages/general-topology/baire-spaces.md.txt @@ -0,0 +1,73 @@ +--- +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. + diff --git a/pages/general-topology/compactness/basics.md b/pages/general-topology/compactness/basics.md new file mode 100644 index 0000000..a1dded7 --- /dev/null +++ b/pages/general-topology/compactness/basics.md @@ -0,0 +1,43 @@ +--- +title: Basics +parent: Compactness +grand_parent: General Topology +nav_order: 1 +published: false +# cspell:words +--- + +# {{ page.title }} of Compact Spaces + +*Compact space* is short for compact topological space. + +{: .definition } +> Suppose $X$ is a topological space. +> A *covering* of $X$ is a collection $\mathcal{A}$ +> of subsets of $X$ such that +> $\bigcup \mathcal{A} = X$. +> A covering $\mathcal{A}$ of $X$ is called *open* +> if each member of the collection $\mathcal{A}$ +> is open in $X$. +> A covering $\mathcal{A}$ is called *finite* +> the collection $\mathcal{A}$ is finite. +> A *subcovering* of a covering $\mathcal{A}$ of $X$ +> is a subcollection $\mathcal{B}$ of $\mathcal{A}$ +> such that $\mathcal{B}$ is a covering of $X$. + +{: .definition } +> A topological space $X$ is called *compact* +> if every open covering of $X$ +> has a finite subcovering. + +{: .theorem } +> Every closed subspace of a compact space is compact. + +{% proof %} +{% endproof %} + +{: .theorem } +> Every compact subspace of a Hausdorff space is closed. + +{% proof %} +{% endproof %} diff --git a/pages/general-topology/compactness/index.md b/pages/general-topology/compactness/index.md new file mode 100644 index 0000000..60c29a0 --- /dev/null +++ b/pages/general-topology/compactness/index.md @@ -0,0 +1,9 @@ +--- +title: Compactness +parent: General Topology +nav_order: 1 +has_children: true +# cspell:words +--- + +# {{ page.title }} diff --git a/pages/general-topology/compactness/tychonoff-product-theorem.md b/pages/general-topology/compactness/tychonoff-product-theorem.md new file mode 100644 index 0000000..2ae78e4 --- /dev/null +++ b/pages/general-topology/compactness/tychonoff-product-theorem.md @@ -0,0 +1,19 @@ +--- +title: Tychonoff Product Theorem +parent: Compactness +grand_parent: General Topology +nav_order: 2 +# cspell:words +--- + +# {{ page.title }} + +{: .theorem-title } +> {{ page.title }} +> {: #{{ page.title | slugify }} } +> +> The product of (an arbitrary family of) compact spaces is compact. + +{% proof %} +TODO +{% endproof %} diff --git a/pages/general-topology/index.md b/pages/general-topology/index.md new file mode 100644 index 0000000..507c29a --- /dev/null +++ b/pages/general-topology/index.md @@ -0,0 +1,11 @@ +--- +title: General Topology +nav_order: 1 +has_children: true +--- + +# {{ page.title }} + +## Recommended Textbooks + +{% bibliography --file general-topology %} diff --git a/pages/general-topology/jordan-curve-theorem.md b/pages/general-topology/jordan-curve-theorem.md new file mode 100644 index 0000000..9da141e --- /dev/null +++ b/pages/general-topology/jordan-curve-theorem.md @@ -0,0 +1,18 @@ +--- +title: Jordan Curve Theorem +parent: General Topology +nav_order: 1 +published: false +# cspell:words +--- + +# {{ page.title }} + +{: .theorem-title } +> {{ page.title }} +> {: #{{ page.title | slugify }} } +> +> ... + +{% proof %} +{% endproof %} -- cgit v1.2.3-54-g00ecf