Locally compact spaces and compactification

Section 1.17 Locally compact spaces and compactification

Definition 1.17.1. Locally compact space.

A topological space \(X\) is locally compact at \(x\in X\) if there is a compact neighborhood of \(x\text{:}\) i.e., if there is a compact set \(K\) and open set \(U\) containing \(x\) such that \(U\subseteq K\text{.}\) The space \(X\) is locally compact if it is locally compact at all points \(x\in X\text{.}\)

Theorem 1.17.2. Local compactness equivalence.

Let \(X\) be a topological space. The following are equivalent.

\(X\) is locally compact Hausdorff.
For all \(x\in X\text{,}\) and for all open sets \(U\) containing \(x\text{,}\) there is an open neighborhood \(V\) of \(x\) such that \(\overline{V}\subseteq U\) and \(\overline{V}\) is compact.

Proof.

It is clear that (2) implies (1): for any \(x\in X\text{,}\) taking \(U=X\text{,}\) we get an open neighborhood \(V\ni x\) such that \(\overline{V}\) is compact. Thus \(K\) is a compact neighborhood of \(x\text{.}\)

We now show (1) implies (2). Fix an \(x\in X\) and open set \(U\) containing \(x\text{.}\) Since \(X\) is locally compact, we have \(x\in U'\subseteq K\) for some open set \(U'\) and compact set \(K\text{.}\) Taking \(U'\cap U\) we may assume further that \(U'\subseteq U\text{.}\) The set \(C=K-U\) is a closed subset of the compact set \(K\text{,}\) hence compact. Since \(X\) is Hausdorff, we can find disjoint open sets \(V', W'\) containing \(x\) and \(C\text{,}\) respectively. (See Theorem 1.14.7.) Let \(V=V'\cap U'\text{.}\) Since \(V\subseteq U'\subseteq K\text{,}\) we have \(\overline{V}\subseteq K\text{;}\) since \(K\) is compact, it follows that \(\overline{V}\) is compact. To show \(\overline{V}\subseteq U\text{,}\) it suffices to show that \(\overline{V}\cap C=K-U\) is empty. Given any \(y\in C=K-U\text{,}\) the open set \(W'\) contains \(y\) and is disjoint from \(V\text{;}\) we conclude that \(x\notin \overline{V}\text{.}\) In all we have shown that for any \(x\in X\) and open set \(U\) containing \(x\text{,}\) there is an open set \(V\) with compact closure \(\overline{V}\) satisfying \(x\in V\subseteq\overline{V}\subseteq U\text{,}\) as desired.

Definition 1.17.3. Embedding.

An embedding between topological spaces \(X\) and \(Y\) is an injective continuous function \(f\colon X\rightarrow Y\) such that \(f\colon X\rightarrow f(X)\) is a homeomorphism (with respect to the subspace topology on \(f(X)\)).

An embedding \(f\colon X\rightarrow Y\) is open (resp., closed) if \(f(X)\) is an open (resp. closed) subset of \(Y\text{:}\) equivalently, if the embedding is an open (resp., a closed) map.

Topological specimen 17. One-point compactification.

Let \((X,\mathcal{T})\) be a topological space, and let \(X^*=X\cup \{p_\infty\}\) be a disjoint union of \(X\) and a single element \(p_\infty\) that is not an element of \(X\text{.}\) The collection

\begin{align*} \mathcal{T}^*\amp=\mathcal{T}\cup \{X^*-K\colon K \subseteq X \text{ closed compact} \}\\ \amp = \mathcal{T}\cup \{(X-K)\cup \{p_\infty\}\colon K\subseteq X \text{ closed compact} \} \end{align*}

defines a topology on \(X^*\) with respect to which the inclusion map

\begin{equation*} i\colon X\rightarrow X^*=X\cup \{p_\infty\} \end{equation*}

is an open immersion. We call \(X^*\) along with the the inclusion map \(i\colon X\rightarrow X^*\) the Alexandroff extension of \(X\text{.}\) When \(X\) is locally compact Hausdorff, we call \(X^*\) the one-point compactification of \(X\text{.}\)

Definition 1.17.4. Compactification.

A compactification of a topological space \(X\) is an embedding \(\iota\colon X\hookrightarrow Y\text{,}\) where \(Y\) is compact and \(\overline{\iota(X)}=Y\text{.}\)

A compactification \(\iota\colon X\hookrightarrow Y\) is a one-point compactification if \(Y-\iota(X)=\{p\}\) is a single point.

Theorem 1.17.5. One-point compactification.

Let \((X,\mathcal{T})\) be a topological space, and let \((X^*,\mathcal{T}^*)\) the Alexandroff extension of \(X\) as defined in Topological specimen 17.

\(\mathcal{T}^*\) is a topology, and the inclusion map \(i\colon X\rightarrow X^*\) is an open immersion with respect to this topology.
\(X^*\) is compact.
The following statements are equivalent.
1. \(\{p_\infty\}\) is open in \(X^*\text{.}\)
2. \(X\) is not dense in \(X^*\)
3. \(X\) is compact.
The following statements are equivalent.
1. \(X^*\) is Hausdorff.
2. \(X\) is locally compact Hausdorff and not compact.

Proof.

Proof of (1).

Assuming \(\mathcal{T}^*\) is a topology, it is easy too see that the inclusion map \(i\colon X\rightarrow X^*\) is both continuous (for open \(U\in \mathcal{T}^*\) we have \(i^{-1}(U)=U\cap X\) open in \(X\)) and an open map, and hence an open immersion.

We show each topology axiom in turn.

Since \(\emptyset\in \mathcal{T}\text{,}\) we have \(\emptyset\in \mathcal{T}^*\text{.}\) Furthermore, since \(\emptyset\) is closed and compact, we have \(X^*=X^*-\emptyset\in \mathcal{T}^*\text{.}\)
Let \(\{U_i\}_{i\in I}\) be a collection of elements of \(\mathcal{T}^*\text{,}\) and let \(U=\bigcup_{i\in I}U_i\text{.}\) If \(U_i\in \mathcal{T}\text{,}\) then \(U\in \mathcal{T}\subseteq \mathcal{T}^*\text{.}\) Suppose \(U_{i_0}=X^*-K\) with \(K\subseteq X\) closed and compact for some \(X\text{.}\) The set \(K'=K-U\) is a closed subset of \(K\) and hence compact. We claim \(U=X^*-K'=\text{,}\) in which case \(U\in \mathcal{T}^*\text{.}\) Indeed, we have

\begin{align*} x\notin X^*-(K-U) \amp \iff x\in K-U \\ \amp \iff x\in K \text{ and } x\notin U\\ \amp \iff x\notin U_{i_0} \text{ and } x\notin U\\ \amp \iff x\notin U\text{.} \end{align*}
Assume \(U_1\) and \(U_2\) are elements of \(\mathcal{T}^*\text{.}\) We have three cases:

\begin{align*} U_1, U_2\in \mathcal{T} \amp \implies U_1\cap U_2\in \mathcal{T}\subseteq \mathcal{T}'\\ U_1\in \mathcal{T}, U_2=X^*-K \amp\implies U_1\cap U_2=U_1-K\in \mathcal{T}\subseteq \mathcal{T}' \\ U_1=X^*-K_1, U_2=X^*-K_2 \amp \implies U_1\cap U_2=X^*-(K_1\cup K_2)\in \mathcal{T}^*\text{,} \end{align*}

where in the last line we’ve used the fact the a finite union of compact sets is compact.

Proof of (2).

Let \(X^*=\bigcup_{i\in I}U_i\) be an open covering. We must have \(p_\infty\in U_{i_0}\) for some \(i_0\in I\text{,}\) in which case \(U_{i_0}=X^*-K\) for some closed compact \(K\subseteq X\text{.}\) Next, we have \(K\subseteq X\cap \bigcup_{i\in I}U_i=\bigcup_{i\in I}V_i\) where \(V_i=X\cap U_i\) is open in \(X\text{.}\) Since \(K\) is compact, there is a finite subcovering \(K\subseteq \bigcup_{j=1}^nV_{i_j}\text{;}\) then we have \(X^*=U_{i_0}\cup \bigcup_{j=1}^nU_{i_j}\text{,}\) a finite subcovering of \(\{U_i\}_{i\in I}\text{.}\)

Proof of (3).

Since \(X^*=X\cup \{p_\infty\}\text{,}\) we see that \(\overline{X}=X\) or \(\overline{X}=X^*\text{.}\) It follows that \(\overline{X}\ne X\) if and only if \(p_\infty\notin \overline{X}\) if and only if \(\overline{X}=X\) if and only if \(\{p_\infty\}\) is open if and only if \(X\) is closed. Now since \(X^*\) is compact, if \(X\) is closed, then \(X\) is compact. Suppose conversely that \(X\) is compact; then by definition of \(\mathcal{T}^*\) the set \(X^*-X=\{p_\infty\}\) is open. The stated equivalences now follow.

Proof of (4).

If \(X^*\) is Hausdorff, then since \(X^*\) is also compact, we see that \(X\) is an open subspace of a locally compact Hausdorff space. The claim now follows from the general fact that an open subspace of a locally compact Hausdorff space is locally compact Hausdorff. This is easy to prove using Theorem 1.17.2, and is left to the reader.

Assume conversely that \(X\) is locally compact Hausdorff. Since \(X\) itself is Hausdorff and is an open subspace of \(X^*\text{,}\) to show \(X^*\) is Hausdorff it suffices to show that given any \(x\in X\text{,}\) there are open disjoint sets \(U,V\in\mathcal{T}^*\) such that \(x\in U\) and \(p_\infty\in V\text{.}\) Since \(X\) is locally compact Hausdorff, there a closed compact neighborhood \(K\) of \(x\text{.}\) It follows that the open sets \(U=K^\circ\in \mathcal{T}\subseteq \mathcal{T}^*\) and \(V=X^*-K\in \mathcal{T}^*\) meet the desired conditions.

Corollary 1.17.6. One-point compactification.

Let \(X\) be locally compact Hausdorff.

If \(X\) is not compact, then the inclusion map

\begin{equation*} i\colon X\rightarrow X^* \end{equation*}

is a one-point compactification of \(X\text{,}\) and \(X^*\) is Hausdorff.
If \(X\) is not compact, and \(\iota\colon X\rightarrow Y\) is a one-point compactification where \(Y\) is Hausdorff, then there exists a unique homeomorphism \(\phi\colon X^*\rightarrow Y\) such that \(\iota\vert_X=\phi\circ i\vert_X\text{.}\)

Corollary 1.17.7. Locally compact Hausdorff spaces.

Let \(X\) be a Hausdorff space. The following statements are equivalent.

\(X\) is locally compact.
\(X\) is a homeomorphic to an open subspace of a compact Hausdorff space.

Proof.

That (1) implies (2) follows from Theorem 1.17.5. For the converse, assume \(X\) is homeomorphic to the open set \(U\subseteq Y\text{,}\) where \(Y\) is compact and Hausdorff. We may use the homeomorphism to identify \(X\) with \(U\text{:}\) thus we think of \(X\subseteq Y\) as an open subspace of \(Y\text{.}\) Now, since \(Y\) is compact, it is locally compact. Hence by Corollary 1.17.7, given any \(x\in X\) we can find an open \(V\ni x\) such that \(\overline{V}\subseteq X\) and \(\overline{V}\) is compact. This shows that every \(x\in X\) has a compact neighborhood \(K=\overline{V}\subseteq X\text{,}\) as desired.

Example 1.17.8. One-point compactification of \(\R\).

We consider \(\R\) with the standard topology.

Produce a one-point compactification of \(\iota\colon (-\pi/2,\pi/2)\hookrightarrow Y\) where \(Y\) is a familiar space.
Produce a one-point compactification of \(\iota\colon \R\hookrightarrow Y\) where \(Y\) is a familiar space.
Use (2) to identify the one-point compactification of \(\Z\) as a subspace of a familiar space.

Solution.

The map \(\iota\colon (-\pi/2,\pi/2)\hookrightarrow S^1\) defined as \(\iota(t)=(\cos 2t, \sin 2t)\) is an open embedding of \((-\pi/2, \pi/2)\) onto \(S^1-\{(-1,0)\}\text{.}\) Thus this is a realization of the one-point compactification of \((-\pi/2, \pi/2)\text{.}\)
Since \(\arctan\colon \R\rightarrow (-\pi/2,\pi/2)\) is a homeomorphism, the map \(\lambda\) defined as the composition

\begin{equation*} \R\xrightarrow{\arctan} (\pi/2,\pi/2)\xrightarrow{\iota} S^1 \end{equation*}

is an open embedding of \(\R\) onto \(S^1-\{(-1,0)\}\text{,}\) providing a realization of the one-point compactification of \(\R\text{.}\) Note that we have by definition \(\lambda(t)=(\cos(2\arctan t), \sin(2\arctan t))\text{.}\)
Since the inclusion \(i\colon \Z\hookrightarrow \R\) is a closed embedding, the restriction

\begin{align*} \lambda\vert_\Z\colon \Z \amp \hookrightarrow S^1\\ n \amp \mapsto (\cos(2\arctan n),\sin(2\arctan n)) \end{align*}

is an embedding (neither open nor closed) of \(\Z\) into \(S^1\text{.}\) Restricting the codomain to the closed (hence compact) subset \(Y=\lambda(\Z)\cup\{(-1,0)\}\) we get a one-point compactification

\begin{equation*} \Z\hookrightarrow \{(-1,0)\}\cup \{(\cos(2\arctan n),\sin(2\arctan n))\colon n\in \Z\}\text{.} \end{equation*}

Topological specimen 18. Stereographic projection.

Let \(P=(0,0,1)\text{,}\) the “north pole” of \(S^2\text{.}\) Given any \(Q=(a,b,c)\in S^2-\{P\}\) the line \(\ell\) passing through \(Q\) and \(P\) intersects the \(xy\)-plane in the unique point \(Q'=(a/(1-c), b/(1-c),0)\text{.}\) The corresponding map

\begin{align*} \phi\colon S^2-\{P\} \amp \rightarrow \R^2\\ (a,b,c) \amp \mapsto (a/(1-c), b/(1-c)) \end{align*}

can easily be shown to be a homeomorphism, with inverse

\begin{align*} \psi\colon \R^2 \amp \mapsto S^2-\{P\}\\ (x,y) \amp \mapsto \left(\frac{2x}{x^2+y^2+1},\frac{2y}{x^2+y^2+1}, \frac{x^2+y^2-1}{x^2+y^2+1}\right)\text{.} \end{align*}

The map \(\phi\) is called the stereographic projection onto \(\R^2\text{.}\) The open embedding \(\psi\colon \R^2\hookrightarrow S^2\) realizes \(S^2\) as the one-point compactification of \(\R^2\text{.}\)

Not surprisingly, this result generalizes easily to \(\R^n\text{.}\) There is a homeomorphism from \(S^n-\{(0,0,\dots, 1)\) to \(\R^n\) mapping \(Q=(a_0,a_1,\dots, a_n)\) to \((a_0/(1-a_n), a_1/(a_n-1),\dots, a_{n-1}/(a_n-1)\text{,}\) whose inverse realizes \(S^n\) as the one-point compactification of \(\R^n\text{.}\)

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