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^[2] 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.

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.^[3] 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

${\displaystyle d_{n}(f,g)=\sup _{x\in K_{n}}\delta {\bigl (}f(x),g(x){\bigr )},\quad f,g\in C(X,M),n\in \mathbb {N} .}$

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

Notes

References

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

External links

• hemicompact space on nLab
• hemicompact on π-Base
  

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This page was last edited on 10 June 2024, at 11:27