Hemicompact space

In mathematics, in the field of topology, a topological space is said to be hemicompact if it has a sequence of compact subsets such that every compact subset of the space lies inside some compact set in the sequence.[1] 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.

Examples

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

Properties

Every hemicompact space is σ-compact and if in addition it is first countable then it is locally compact.

Applications

If ${\displaystyle X}$ is a hemicompact space, then the space ${\displaystyle C(X,M)}$ of all continuous functions ${\displaystyle f:X\to M}$ to a metric space ${\displaystyle (M,\delta )}$ with the compact-open topology is metrizable.[2] To see this, take a sequence ${\displaystyle K_{1},K_{2},\dots }$ of compact subsets of ${\displaystyle X}$ such that every compact subset of ${\displaystyle X}$ lies inside some compact set in this sequence (the existence of such a sequence follows from the hemicompactness of ${\displaystyle X}$). Define pseudometrics

${\displaystyle d_{n}(f,g)=\sup _{x\in K_{n}}\left|(f(x)-g(x))\right|,\quad f,g\in C(X,M),n\in \mathbb {N} .}$

Then

${\displaystyle d(f,g)=\sum _{n=1}^{\infty }{\frac {1}{2^{n}}}\cdot {\frac {d_{n}(f,g)}{1+d_{n}(f,g)}}}$

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

See also

• Compact space
• Locally compact space
• Lindelöf space

Notes

1. Willard 2004, Problem set in section 17.
2. Conway 1990, Example IV.2.2.

References

• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.CS1 maint: ref=harv (link)
• Conway, J. B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. 96. Springer Verlag. ISBN 0-387-97245-5.CS1 maint: ref=harv (link)