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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/basics.md | |
parent | 777f9d3fd8caf56e6bc6999a4b05379307d0733f (diff) | |
download | site-28407333ffceca9b99fae721c30e8ae146a863da.tar.zst |
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Diffstat (limited to 'pages/general-topology/compactness/basics.md')
-rw-r--r-- | pages/general-topology/compactness/basics.md | 51 |
1 files changed, 27 insertions, 24 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 %} |