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In Euclidean space, a subset is compact if and only if it is closed and bounded.

A space is limit point compact if every infinite subset has a limit point.

If $f: X \rightarrow Y$ is continuous and $X$ is compact, then $f(X)$ is compact.

If a covering space is compact, its base space is compact as well.

Every bounded sequence in $\mathbb{R}^n$ has a convergent subsequence.

The product of any collection of compact spaces is compact in the product topology.

The process or result of making a non-compact space into a compact space by adding 'points at infinity' or otherwise extending its topology.

A space is locally compact if every point has a compact neighborhood.

A topological space is compact if every open cover has a finite subcover.

Projection maps from a compact product space to its factors are open and continuous.

A topology on the set of continuous functions between two topological spaces where the basic open sets are defined by compact subsets of the domain and open subsets of the codomain.

In a Hausdorff space, every compact subset is closed.

A topological space is sequentially compact if every sequence has a convergent subsequence.

A one-point compactification of a non-compact topological space that adds a single 'point at infinity' to make it compact.