A compact space is a fundamental concept in topology, a branch of mathematics. A space is considered compact if every open cover has a finite subcover. This means that if you can cover the space with a collection of open sets, you can always find a limited number of those sets that still cover the entire space. Compactness is an important property because it often allows for the extension of various theorems. For example, in real analysis, the Heine-Borel theorem states that a subset of Euclidean space is compact if and only if it is closed and bounded. This makes compact spaces useful in many areas of mathematics.