*Definition [#u31e7fc3]
A topological space is said to be hemicompact if there is a sequence of compact subsets (called admissible sequence) such that every compact subset is contained in some member of the sequence.

*Property [#bf59614c]
-A hemicompact space is the union of the admissible sequence since every singleton in the space is a compact set.
-Every first countable hemicompact space is [[locally compact]].
-If X is hemicompact, then the space C(X) of all continuous functions from X to the real is metrizable. The metric is given by 
#ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5bd_n%28f%2cg%29=%5csup_%7bx%5cin%20K_n%7d%7cf%28x%29-g%28x%29%7c%2c%5cquad%20d%28f%2cg%29=%5cfrac%7b1%7d%7b2%5en%7d%5cfrac%7bd_n%28f%2cg%29%7d%7b1+d_n%28f%2cg%29%7d%5c%5d%7d%25.png);
where K_n are the admissible sequence.
*Reference [#fc087824]
Willard, Stephen, ''General Topology'', Dover (2004).