Hemicompact space explained

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. 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^[1] 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

is a hemicompact space, then the space

C(X,M)

of all continuous functions

f:X\toM

to a metric space

(M,\delta)

with the compact-open topology is metrizable. To see this, take a sequence

K_1,K_2,...

of compact subsets of

such that every compact subset of

lies inside some compact set in this sequence (the existence of such a sequence follows from the hemicompactness of

). Define pseudometrics

d_n(f,g)=

\sup
	x\inK_n

\deltal(f(x),g(x)r), f,g\inC(X,M),n\inN.

Then

d(f,g)=

	infty
\sum
	n=1

	1
	2ⁿ

⋅

	d_n(f,g)
	1+d_n(f,g)

defines a metric on

C(X,M)

which induces the compact-open topology.

References

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

External links

hemicompact space on nLab
hemicompact on π-Base

Notes and References

Willard 2004, p. 126