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

In a Hausdorff space, every compact subset is closed.

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

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

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.

For every open cover of a compact metric space, there exists a positive number $\delta$ such that any subset of diameter less than $\delta$ is contained in some member of the cover.

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

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

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

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

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

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

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

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