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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
commit | 28407333ffceca9b99fae721c30e8ae146a863da (patch) | |
tree | 67fa2b79d5c48b50d4e394858af79c88c1447e51 /pages/general-topology/compactness | |
parent | 777f9d3fd8caf56e6bc6999a4b05379307d0733f (diff) | |
download | site-28407333ffceca9b99fae721c30e8ae146a863da.tar.zst |
Update
Diffstat (limited to 'pages/general-topology/compactness')
-rw-r--r-- | pages/general-topology/compactness/basics.md | 51 | ||||
-rw-r--r-- | pages/general-topology/compactness/index.md | 41 | ||||
-rw-r--r-- | pages/general-topology/compactness/tychonoff-product-theorem.md | 12 |
3 files changed, 72 insertions, 32 deletions
diff --git a/pages/general-topology/compactness/basics.md b/pages/general-topology/compactness/basics.md index a1dded7..c7249ff 100644 --- a/pages/general-topology/compactness/basics.md +++ b/pages/general-topology/compactness/basics.md @@ -4,40 +4,43 @@ 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. +{% 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 $X$. +{% enddefinition %} + +{% definition %} +A topological space $X$ is called *compact* +if every open covering of $X$ +has a finite subcovering. +{% enddefinition %} + +{% theorem %} +Every closed subspace of a compact space is compact. +{% endtheorem %} {% proof %} {% endproof %} -{: .theorem } -> Every compact subspace of a Hausdorff space is closed. +{% theorem %} +Every compact subspace of a Hausdorff space is closed. +{% endtheorem %} {% proof %} {% endproof %} diff --git a/pages/general-topology/compactness/index.md b/pages/general-topology/compactness/index.md index 60c29a0..37e9b4d 100644 --- a/pages/general-topology/compactness/index.md +++ b/pages/general-topology/compactness/index.md @@ -1,9 +1,46 @@ --- title: Compactness parent: General Topology -nav_order: 1 +nav_order: 5 has_children: true -# cspell:words +has_toc: false --- # {{ page.title }} + +## Compactness in Terms of Closed Sets + +{% theorem %} +A topological space $X$ is compact +if and only if it has the following property: +- Given any collection $\mathcal{C}$ of closed subsets of $X$, + if every finite subcollection of $\mathcal{C}$ has nonempty intersection, + then $\mathcal{C}$ has nonempty intersection. +{% endtheorem %} + +{% proof %} +By definition, a topological space $X$ is compact +if and only if it has the following property: +- Given any collection $\mathcal{O}$ of open subsets of $X$, + if $\mathcal{O}$ covers $X$, + then there exists a finite subcollection of $\mathcal{O}$ that covers $X$. + +If $\mathcal{A}$ is a collection of subsets of $X$, +let $\mathcal{A}^c = \braces{ X \setminus A : A \in \mathcal{A}}$ denote the collection of the complements of its members. +Clearly, $\mathcal{B}$ is a subcollection of $\mathcal{A}$ +if and only if $\mathcal{B}^c$ is a subcollection of $\mathcal{A}^c$. +Moreover, note that $\mathcal{B}$ covers $X$ if and only if +$\mathcal{B}^c$ has empty intersection. +Taking the contrapositive, we reformulate above property: +- Given any collection $\mathcal{O}$ of open subsets of $X$, + if every finite subcollection of $\mathcal{O}^c$ has nonempty intersection, + then $\mathcal{O}^c$ has nonempty intersection. + +To complete the proof, observe that a collection $\mathcal{A}$ consists of open subsets of $X$ +if and only if $\mathcal{A}^c$ consists of closed subsets of $X$. +{% endproof %} + +{% definition Finite Intersection Property%} +TODO +{% enddefinition %} + diff --git a/pages/general-topology/compactness/tychonoff-product-theorem.md b/pages/general-topology/compactness/tychonoff-product-theorem.md index 2ae78e4..ddf3800 100644 --- a/pages/general-topology/compactness/tychonoff-product-theorem.md +++ b/pages/general-topology/compactness/tychonoff-product-theorem.md @@ -3,17 +3,17 @@ 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. +{% theorem * Tychonoff Product Theorem %} +The product of (an arbitrary family of) compact spaces is compact. +{% endtheorem %} {% proof %} TODO {% endproof %} + +Important Application: +[Banach–Alaoglu Theorem](/pages/functional-analysis-basics/banach-alaoglu-theorem.html). |