Hemicompact space

In mathematics, in the field of topology, a Hausdorff 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] 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