Local Compactness and One-Point Compactification

Many important spaces (such as $\mathbb{R}^n$ ) are not compact but are "locally compact" -- every point has a compact neighborhood. The one-point compactification provides a canonical way to compactify such spaces by adding a single point at infinity.

Local Compactness

Definition5.8Locally Compact Space

A topological space $X$ is locally compact if every point $x \in X$ has a compact neighborhood. That is, for each $x \in X$ , there exists an open set $U$ and a compact set $K$ with $x \in U \subseteq K$ .

If $X$ is Hausdorff, this is equivalent to: every point has an open neighborhood whose closure is compact.

ExampleLocally Compact Spaces

$\mathbb{R}^n$ is locally compact: each point $x$ lies in the open ball $B(x, 1)$ whose closure $\overline{B}(x, 1)$ is compact.
Any compact space is locally compact (take $K = X$ ).
Discrete spaces are locally compact (singletons are compact neighborhoods).
$\mathbb{Q}$ is not locally compact: no point of $\mathbb{Q}$ has a compact neighborhood. (If $U$ is an open neighborhood of $q \in \mathbb{Q}$ , then $\overline{U}^{\mathbb{Q}}$ is not compact: every compact subset of $\mathbb{Q}$ has empty interior.)
$\ell^2$ (infinite-dimensional Hilbert space) is not locally compact.

Theorem5.5Properties of Locally Compact Hausdorff Spaces

Let $X$ be a locally compact Hausdorff space.

Every open subset of $X$ is locally compact.
Every closed subset of $X$ is locally compact.
If $U$ is open and $x \in U$ , there exists an open set $V$ with $x \in V \subseteq \overline{V} \subseteq U$ and $\overline{V}$ compact.

Proof

(3): By local compactness, there is an open $W$ with $x \in W$ and $\overline{W}$ compact. Consider $U \cap W$ , an open neighborhood of $x$ in the compact Hausdorff space $\overline{W}$ . Since compact Hausdorff spaces are regular, there exists an open $V'$ in $\overline{W}$ with $x \in V' \subseteq \overline{V'}^{\overline{W}} \subseteq U \cap W$ . Let $V$ be the interior of $V'$ in $X$ ; then $\overline{V}$ is closed in $\overline{W}$ , hence compact, and $\overline{V} \subseteq U$ .

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One-Point Compactification

Definition5.9One-Point (Alexandroff) Compactification

Let $X$ be a topological space. The one-point compactification of $X$ is the space $X^* = X \cup \{\infty\}$ (where $\infty$ is a point not in $X$ ) with the topology: $\tau^* = \tau_X \cup \{(X \setminus K) \cup \{\infty\} : K \subseteq X \text{ is compact and closed}\}.$

That is, the open sets are the open sets of $X$ together with complements of closed compact subsets of $X$ (with $\infty$ added).

Theorem5.6Properties of the One-Point Compactification

Let $X$ be a topological space and $X^* = X \cup \{\infty\}$ its one-point compactification.

$X^*$ is compact.
The inclusion $X \hookrightarrow X^*$ is an embedding (the subspace topology on $X$ in $X^*$ is the original topology).
$X^*$ is Hausdorff if and only if $X$ is locally compact and Hausdorff.
If $X$ is already compact, then $\{\infty\}$ is an isolated point (clopen) in $X^*$ .

Proof

(1): Let $\{U_\alpha\}$ be an open cover of $X^*$ . Some $U_{\alpha_0}$ contains $\infty$ , so $U_{\alpha_0} = (X \setminus K) \cup \{\infty\}$ for some closed compact $K \subseteq X$ . The remaining sets $\{U_\alpha\}_{\alpha \neq \alpha_0}$ cover $K$ (which lies in $X$ ). Since $K$ is compact, finitely many $U_{\alpha_1}, \ldots, U_{\alpha_n}$ cover $K$ . Then $\{U_{\alpha_0}, U_{\alpha_1}, \ldots, U_{\alpha_n}\}$ is a finite subcover of $X^*$ .

(3, $\Leftarrow$ ): Suppose $X$ is locally compact Hausdorff. To separate $x \in X$ from $\infty$ : by local compactness, there is an open $U$ with $\overline{U}$ compact. Then $U$ is an open neighborhood of $x$ and $(X \setminus \overline{U}) \cup \{\infty\}$ is an open neighborhood of $\infty$ , and they are disjoint. Separation of two points in $X$ follows from Hausdorff of $X$ .

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Classical Examples

ExampleOne-Point Compactifications

$\mathbb{R}^n$ : The one-point compactification of $\mathbb{R}^n$ is homeomorphic to $S^n$ . For $n = 1$ , this gives $\mathbb{R}^* \cong S^1$ (the circle). For $n = 2$ , $\mathbb{R}^2{}^* \cong S^2$ (the Riemann sphere). The homeomorphism is given by inverse stereographic projection.
$\mathbb{C}$ : $\mathbb{C}^* = \mathbb{C} \cup \{\infty\} \cong S^2$ is the Riemann sphere, fundamental in complex analysis.
$\mathbb{Z}$ : The one-point compactification $\mathbb{Z}^* = \mathbb{Z} \cup \{\infty\}$ is homeomorphic to $\{0\} \cup \{1/n : n \in \mathbb{Z}^+\}$ (a convergent sequence with its limit).
$(0, 1)$ : $((0,1))^* \cong S^1$ , since $(0,1) \cong \mathbb{R}$ .

RemarkUniqueness of Compactification

For a locally compact Hausdorff space $X$ , the one-point compactification $X^*$ is the unique (up to homeomorphism) compact Hausdorff space containing $X$ as an open dense subspace with a one-point complement. However, there are other compactifications: the Stone--Cech compactification $\beta X$ is the maximal one (see Chapter 8).