Compactness

Compactness turns a possibly infinite covering problem into a finite one.

If $X=\bigcup_{i\in I}U_i$ with each $U_i$ open, compactness says that there are finitely many indices $i_1,\ldots,i_n$ for which

A useful consequence

Continuous Images of Compact Spaces explains why compactness survives continuous maps. This is one reason the definition is so useful beyond Euclidean space.