3.13 Locally Compact Hausdorff Spaces
Definition 3.13.1 (Locally Compact Hausdorff Space).label Let $X$ be a Hausdorff space, then the following are equivalent:
- (1)
For any $x\in X$, there exists $K\in\cali{N}(x)$ is compact.
- (2)
For any $x\in X$, $\cali{N}(x)$ admits a fundamental system of neighborhoods consisting of compact sets.
- (3)
For any $x\in X$, $\cali{N}(x)$ admits a fundamental system of neighborhoods consisting of precompact sets.
If the above holds, then $X$ is a locally compact Hausdorff space
Proof. (1)$\implies$(2) Let $K\in\cali{N}(x)$ be compact and $U\in\cali{N}^{o}(x)$ then there exists open sets $V\in\cali{N}(x)$ such that $V\suf K$ so $V\cap K\in\cali{N}^{o}(x)$ in the compact Hausdorff space $K$. By Proposition 3.10.6 there exists an open set $W\suf K$ satisfying
where $\cl{W}$ is unambigous by Proposition 3.13.2. Since $K$ is compact, so is $\cl{W}$.
(2)$\implies$(3) Let $U\in\cali{N}(x)$, then there exists $K\in\cali{N}(x)$ compact such that $x\in K\suf U$. Since $X$ is Hausdorff, $K$ is closed so $\cl{K^o}\suf K$ is compact.
For any $x\in X$, $\cali{N}(x)\neq\emp$ hence (2)/(3)$\implies$(1).$\square$
Proposition 3.13.2.label Let $X$ be a topological space, $Y\suf X$ be a subspace then for any subset $A\suf X$
In particular if $Y$ is closed in $X$ and $A\suf Y$ then
Proof. Observe
$\square$
Lemma 3.13.3.label Let $X$ be a LCH space, $K\suf X$ be compact, and $U\in\cali{N}(K)$, then there exists $V\in\cali{N}^{o}(K)$ precompact such that $K\suf V\suf\cl{V}\suf U$. In particular $X$ is regular.
Proof. For each $x\in K$, there exists $V_{x}\in\cali{N}^{o}(x)$ such that $x\in V_{x}\suf\cl{V_x}\suf U$ by Definition 3.13.1(3). Since $K$ is compact there exists $\curl{x_j}_{1}^{n}\suf K$ such that $K\suf \cups{n}{j=1}V_{x_j}$ hence
we can take $V=\cups{n}{j=1}V_{x_j}\in\cali{N}^{o}(K)$.$\square$
Theorem 3.13.4 (Urysohn’s Lemma (LCH)).label Let $X$ be a LCH space, $K\suf X$ be compact, $U\in\cali{N}(K)$, and $V$ as in Lemma 3.13.3 then there exists $F\in C_{c}(X;[0,1])$ such that $F|_{K}= 1,\supp{F}\suf \cl{V}$.
Proof. By Lemma 3.13.3, there exists $V,W\in\cali{N}^{o}(K)$ precompact such that
As $\cl{W}$ is compact, it is normal by Proposition 3.10.6. Since $X$ is Hausdorff, $K\suf \cl{W}$ is closed.
By Urysohn’s Lemma, there exists $f\in C(\cl{W};[0,1])$ such that $f|_{K}=1$ and $f|_{\cl{W}\del V}=0$. Let
then $F|_{K}=1,\supp{F}\suf\supp{f}\suf \cl{V}$ and by Gluing Lemma for Continuous Functions $F\in C_{c}(X;[0,1])$.$\square$
Theorem 3.13.5 (Tietze Extension Theorem (LCH)).label Let $X$ be a LCH space, $K\suf X$ compact, $U\in\cali{N}^{o}(K)$, $f\in C(K;\R)$, and $V$ as in Lemma 3.13.3 then there exists $F\in C_{c}(X;\R)$ such that $F|_{K}=f,\supp{F}\suf \cl{V}$ .
Proof. By Lemma 3.13.3, there exists $V,W\in\cali{N}^{o}(K)$ precompact such that
As $\cl{W}$ is compact, it is normal by Proposition 3.10.6. Since $X$ is Hausdorff, $K\suf \cl{W}$ is closed.
By Tietze Extension Theorem, there exists $g\in C(\cl{W};\R)$ such that $g|_{K}=f$. By Urysohn’s Lemma, there exists $\eta\in C_{c}(X;[0,1])$ such that $\eta|_{K}=1,\supp{\eta}\suf \cl{V}$. Let
then $F|_{K}=g|_{K}=f,\supp{F}\suf\supp{\eta}\suf \cl{V}$ and by Gluing Lemma for Continuous Functions $F\in C_{c}(X;\R)$.$\square$
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