Heine-Borel Theorem

The Heine-Borel theorem is an important theorem in elementary topology.

Statement

Let $E$ be any subset of $\mathbb R^n$ . Then $E$ is compact if and only if $E$ is closed and bounded.

This statement does not hold if $\mathbb R^n$ is replaced by an arbitrary metric space $X$ . However, a modified version of the theorem does hold:

Let $X$ be any metric space, and let $E$ be a subset of $X$ . Then $E$ is compact if and only if $E$ is closed and totally bounded.

In $\mathbb R^n$ the totally bounded sets are precisely the bounded sets, so this new formulation does indeed imply the original theorem.

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