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. If a hemicompact space is weakly locally compact, then it is exhaustible by compact sets.
Applications
If [math]\displaystyle{ X }[/math] is a hemicompact space, then the space [math]\displaystyle{ C(X, M) }[/math] of all continuous functions [math]\displaystyle{ f : X \to M }[/math] to a metric space [math]\displaystyle{ (M, \delta) }[/math] with the compact-open topology is metrizable.[2] To see this, take a sequence [math]\displaystyle{ K_1,K_2,\dots }[/math] of compact subsets of [math]\displaystyle{ X }[/math] such that every compact subset of [math]\displaystyle{ X }[/math] lies inside some compact set in this sequence (the existence of such a sequence follows from the hemicompactness of [math]\displaystyle{ X }[/math]). Define pseudometrics
- [math]\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}. }[/math]
Then
- [math]\displaystyle{ d(f,g) = \sum_{n=1}^{\infty} \frac{1}{2^n} \cdot \frac{d_n (f,g)}{1+d_n (f,g)} }[/math]
defines a metric on [math]\displaystyle{ C(X,M) }[/math] which induces the compact-open topology.
See also
- Compact space
- Exhaustible by compact sets
- Locally compact space
- Lindelöf space
Notes
- ↑ Willard 2004, Problem set in section 17.
- ↑ Conway 1990, Example IV.2.2.
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. 96. Springer Verlag. ISBN 0-387-97245-5.
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