Hemicompact space#Applications

• Every compact space is hemicompact.
• The real line is hemicompact.
• Every locally compact Lindelöf space is hemicompact.

Every hemicompact space is σ-compactWillard 2004, p. 126 and if in addition it is first countable then it is locally compact. If a hemicompact space is weakly locally compact, then it is exhaustible by compact sets.

If $X$ is a hemicompact space, then the space $C(X,\; M)$ of all continuous functions $f\; :\; X\; \backslash to\; M$ to a metric space $(M,\; \backslash delta)$ with the compact-open topology is metrizable.{{harvnb|Conway|1990|loc=Example IV.2.2.}} To see this, take a sequence $K\_1,K\_2,\backslash dots$ of compact subsets of $X$ such that every compact subset of $X$ lies inside some compact set in this sequence (the existence of such a sequence follows from the hemicompactness of $X$). Define pseudometrics

:$d\_n\; (f,g)\; =\; \backslash sup\_\{x\; \backslash in\; K\_n\}\; \backslash delta\backslash bigl(\; f(x),\; g(x)\; \backslash bigr),\; \backslash quad\; f,g\; \backslash in\; C(X,M),\; n\; \backslash in\; \backslash mathbb\{N\}.$

Then

:$d(f,g)\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{1\}\{2^n\}\; \backslash cdot\; \backslash frac\{d\_n\; (f,g)\}\{1+d\_n\; (f,g)\}$

defines a metric on $C(X,M)$ which induces the compact-open topology.

• Compact space
• Exhaustible by compact sets
• Locally compact space
• Lindelöf space

{{reflist}}

• {{cite book | last=Willard|first=Stephen | title=General Topology | publisher=Dover Publications | year=2004 | isbn=0-486-43479-6}}
• {{cite book|last=Conway|first=J. B.|author-link=John B. Conway|title=A Course in Functional Analysis|year=1990|series=Graduate Texts in Mathematics|volume=96|publisher=Springer Verlag|isbn=0-387-97245-5}}

• hemicompact space on nLab
• [https://topology.pi-base.org/properties/P000111 hemicompact] on π-Base

Category:Compactness (mathematics)

Category:Properties of topological spaces

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