What is locally compact space?

$X$ is locally compact if it is Hausdorff and every point has a compact neighborhood ($\R^n$; open subsets of $\R^n$; discrete spaces — but not $\Q$, see Exercise 6.9). Every locally compact space embeds in a compact one: the one-point compactification $\hat X = X \cup \{\infty\}$, whose opens are those of $X$ together with the complements (in $\hat X$) of compact subsets of $X$. One checks the axioms directly; $\hat X$ is compact (a cover has a member containing $\infty$, whose complement is compact, covered by finitely many others) and Hausdorff (separate $x$ from $\infty$ by a compact neighborhood of $x$ and its complement). Example: $\hat\R \cong S^1$, and $\widehat{\R^n} \cong S^n$ by stereographic projection (Exercise 6.11).