(mathematics)
(a) Set S is
compact if every
open cover of S can be reduced to a finite open cover.
(b) Set S is
sequentially compact if every sequence has a
convergent subsequence.
(c) Set S is
countably compact if every infinite set has a
limit point.
In
metric spaces, (a), (b), and (c) are equivalent, while in general
topology this is not always the case.
The
Heine-Borel Theorem explains more about what properties are associated with compact sets, based on the definition of compact.
Updated 20 June 2002: Revised for clarity.
Source: Set Theory and Metric Spaces, by Irving Kaplansky. ISBN 0-8284-0298-1