Local compactness says compact neighborhoods exist around points. It is the condition that lets a noncompact Hausdorff space acquire a well behaved point at infinity.
Local compactness
There are several equivalent formulations in Hausdorff spaces.
Definition. A space \(X\) is locally compact at \(x\) if \(x\) has a neighborhood whose closure is compact, or equivalently in Hausdorff spaces, if \(x\) has a compact neighborhood. The space is locally compact if this holds at every point.
Proposition. If \(X\) is locally compact Hausdorff and \(U\) is an open neighborhood of \(x\), then there is an open set \(V\) such that \(x\in V\subseteq\overline V\subseteq U\) and \(\overline V\) is compact.
Proof. Choose a compact neighborhood \(K\) of \(x\) and an open \(W\) with \(x\in W\subseteq K\). Inside the compact Hausdorff subspace \(K\), regularity separates \(x\) from the closed set \(K\setminus(U\cap W)\). This gives an open \(V\) whose closure in \(K\), hence in \(X\), is compact and contained in \(U\).
The point at infinity
The one-point compactification declares complements of compact sets to be neighborhoods of infinity.
Definition. For a noncompact locally compact Hausdorff space \(X\), set \(X^\ast=X\cup\{\infty\}\). The open sets are the open subsets of \(X\), together with sets of the form \(X^\ast\setminus K\), where \(K\subseteq X\) is compact and closed.
Theorem. With this topology, \(X^\ast\) is compact Hausdorff and \(X\) is a dense open subspace of \(X^\ast\).
Proof. An open cover of \(X^\ast\) contains a neighborhood \(X^\ast\setminus K\) of \(\infty\); the remaining compact set \(K\) is covered by finitely many other cover elements, so the whole cover has a finite subcover. Hausdorff separation between points of \(X\) is inherited from \(X\). To separate \(x\in X\) from \(\infty\), choose a compact neighborhood \(K\) of \(x\); its interior contains \(x\), and \(X^\ast\setminus K\) contains \(\infty\). Density follows because every neighborhood of \(\infty\) meets \(X\) when \(X\) is noncompact.
Characterization
The construction is not arbitrary: it characterizes locally compact Hausdorff spaces among compact spaces with one point removed.
Proposition. If \(Y\) is compact Hausdorff and \(p\in Y\), then \(Y\setminus\{p\}\) is locally compact Hausdorff.
Proof. Hausdorffness is inherited by subspaces. If \(x\ne p\), choose disjoint open neighborhoods of \(x\) and \(p\); by regularity of compact Hausdorff spaces, choose \(V\) with \(x\in V\) and \(\overline V\) compact contained in \(Y\setminus\{p\}\). Thus \(V\cap(Y\setminus\{p\})\) has compact closure in the subspace.
Corollary. A noncompact Hausdorff space admits a Hausdorff one-point compactification exactly when it is locally compact.
Proof. The forward direction follows from the proposition applied to the compactification. The reverse direction is the construction of \(X^\ast\).
Proper maps
Local compactness turns compactness at infinity into a useful mapping condition.
Definition. A continuous map \(f:X\to Y\) between locally compact Hausdorff spaces is proper if \(f^{-1}(K)\) is compact for every compact \(K\subseteq Y\).
Proposition. A continuous map \(f:X\to Y\) is proper if and only if it extends continuously to \(f^\ast:X^\ast\to Y^\ast\) with \(f^\ast(\infty)=\infty\).
Proof. Continuity at points of \(X\) is unchanged. Continuity at \(\infty\) says that for each compact \(K\subseteq Y\), the inverse image of \(Y^\ast\setminus K\) is a neighborhood of \(\infty\), meaning \(X\setminus f^{-1}(K)\) contains the complement of some compact subset of \(X\). This is equivalent to \(f^{-1}(K)\) being compact.
Uniqueness and failure of Hausdorffness
Theorem. A noncompact Hausdorff space \(X\) has a Hausdorff one-point compactification if and only if it is locally compact. When it exists, it is unique up to a homeomorphism fixing \(X\).
Proof. If \(X\) is locally compact, adjoin \(\infty\) and declare its neighborhoods to be \[ \{\infty\}\cup(X\setminus K),\qquad K\subset X\text{ compact}. \] Local compactness separates \(x\in X\) from \(\infty\), and any open cover contains one such neighborhood, leaving only a compact remainder. Conversely, if \(X\cup\{\infty\}\) is compact Hausdorff, separate \(x\) and \(\infty\) by neighborhoods whose closures are disjoint; the closure of the \(x\)-neighborhood is compact and contained in \(X\). The compact subsets of \(X\) are exactly the closed subsets missing \(\infty\), so any two such compactifications have the same topology.
Example. \[ (\mathbb R^n)^+\cong S^n \] by stereographic projection. More generally, \[ (X\times\mathbb R)^+ \] is closely related to the suspension of \(X^+\), making one-point compactification a bridge from noncompact geometry to based topology.