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| In [[mathematics]], in the field of [[topology]], a [[topological space]] is said to be '''hemicompact''' if it has a sequence of [[compact space|compact]] subsets such that every compact subset of the space lies inside some compact set in the sequence. Clearly, this forces the union of the sequence to be the whole space, because every point is compact and hence must lie in one of the compact sets.
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| ==Examples==
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| * Every [[compact space]] is hemicompact.
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| * The [[real line]] is hemicompact.
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| * Every locally compact [[Lindelöf space]] is hemicompact.
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| ==Properties==
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| Every [[first-countable space|first countable]] hemicompact space is [[locally compact space|locally compact]].
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| If <math>X</math> is a hemicompact space, then the space <math>C(X, M)</math> of all continuous functions <math>f : X \to M</math> to a [[metric space]] <math>(M, \delta)</math> with the [[compact-open topology]] is [[Metrization theorem|metrizable]]. To see this, take a sequence <math>K_1,K_2,\dots</math> of compact subsets of <math>X</math> such that every compact subset of <math>X</math> lies inside some compact set in this sequence (the existence of such a sequence follows from the hemicompactness of <math>X</math>). Denote
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| :<math>d_n (f,g) = \sup_{x \in K_n} \delta(f(x), g(x))</math>
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| for <math>f,g \in C(X,M)</math> and <math>n \in \mathbb{N}</math>. Then
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| :<math>d(f,g) = \sum_{n=1}^{\infty} \frac{1}{2^n} \cdot \frac{d_n (f,g)}{1+d_n (f,g)}</math>
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| defines a metric on <math>C(X,M)</math> which induces the compact-open topology.
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| ==See also==
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| * [[Compact space]]
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| * [[Locally compact space]]
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| * [[Lindelöf space]]
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| ==References==
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| *{{cite book | author=Willard, Stephen | title=General Topology | publisher=Dover Publications | year=2004 | id=ISBN 0-486-43479-6}}
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| {{topology-stub}}
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| [[Category:Compactness (mathematics)]]
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| [[Category:Properties of topological spaces]]
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