3.13.6 Paracompactness and LCH Spaces

Proof. (1)$\implies$(2) Let $\curl{K_n}_{1}^{\infty}\suf\cali{P}(X)$ be a covering of $X$ by compact sets and $U_{0}=\emp$. Suppose inducitvely we have constructed $\curl{U_i}_{0}^{n}$ satisfying:

1. (a)

   For each $0\leq j\leq n,U_{j}$ is precompact open set.

2. (b)

   For each $0\leq j<n,\cl{U_j}\suf U_{j+1}$.

3. (c)

   For each $1\leq j\leq n, U_{j}\supf \cups{j}{i=1}K_{i}$.

Pick $U_{n+1}\in \cali{N}^{o}(\cl{U}_{n}\cup K_{n+1})$ precompact by Lemma 3.13.3 then by (c)

Thus $\curl{U_i}_{0}^{n+1}$ satisfies (a,b,c) and $\curl{U_n}_{1}^{\infty}$ is an exhaustion of $X$ by compact sets. (2)$\implies$(1) follows by definition.$\square$

Proof. $\square$

Proof. For each $E\in\cali{E},\curl{F\in\cali{E}:F\cap\cl{E}\neq\emp}$ is finite by Lemma 3.12.2. Let

then $F_{E}\in\cali{N}(\cl{E})$ is precompact and open. Let $N\suf X$ and $E\in\cali{E}$. If $N\cap F_{E}\neq\emp$, then there exists $F\in\cali{E}$ such that $N\cap F\neq\emp$ and $F\cap\cl{E}\neq\emp$. Thus

By Lemma 3.12.3, $\curl{\cl{E}:E\in\cali{E}}$ is also locally finite. Hence for each $x\in X$, there exists $N\in\cali{N}(x)$ such that $\curl{F\in\cali{E}:N\cap F\neq\emp}$ is finite so $\curl{F_E}_{E\in\cali{E}}$ is locally finite.

For each $x\in X$, there exists $E\in\cali{E}$ and $N_{x}\in\cali{N}^{o}(x)$ precompact such that $x\in N_{x}\suf\cl{N_x}\suf E$. By compactness of $\cl{E}$ there exists $X_{E}\suf X$ finite such that

1. (a)

   $\curl{N_x:x\in X_E}$ is a covering of $\cl{E}$

2. (b)

   For every $x\in X_{E},N_{x}\cap E\neq\emp$

Let $X_{\cali{E}}=\cups{}{E\in\cali{E}}X_{E}$ and for each $E\in\cali{E}$ let

then $\curl{G_E}_{E\in\cali{E}}$ is an open cover of $X$. Since $G_{E}\suf E$ for all $E\in\cali{E},\curl{G_E }_{E\in\cali{E}}$ is locally finite. Let $F\in\cali{E},x\in X_{F}$ then $N_{x}\cap F\neq\emp$. Fix $E\in\cali{E}$, if $N_{x}\suf E$ then $E\cap F\neq\emp$ thus

is finite by Lemma 3.12.2, so

$\square$

Proof. (1)$\implies$(2): For each $x\in X$, there exists a precompact open neighborhood $U_{x}\in \cali{N}^{o}(x)$. Since $\curl{U_x}_{x\in X}$ is an open cover of $X$, there exists a locally finite refinement $\cali{V}$. For each $V\in\cali{V}$, there exists $x\in X$ such that $V\suf U_{x}$ hence $\cl{V}\suf \cl{U_x}$ is compact. (2)$\implies$(3): Let $\cali{F}\suf\cali{P}(X)$ be a locally finite open cover of $X$ consisting of precompact open sets. By Lemma 3.13.9, there exists a locally finite open cover $\curl{G_F }_{F\in\cali{F}}$ of $X$ consisting of precompact open sets such that $\cl{F}\suf G_{F}$ for all $F\in\cali{F}$. For each $F\in\cali{F}$, let

then $\cali{U}_{F}$ is a precompact open cover of $\cl{F}$. Pick $\cali{V}_{F}\suf \cali{U}_{F}$ by compactness of $\cl{F}$ then $\cali{V}=\cups{}{F\in\cali{F}}\cali{V}_{F}$ is a precompact open cover of $X$. For each $x\in X$, there exists $N_{x}\in\cali{N}(x)$ such that $\curl{F\in\cali{F}:N_x\cap G_F\neq\emp}$ is finite thus

is finite and $\cali{V}$ is locally finite. (3)$\implies$(4): By Lemma 3.13.9 (4)$\implies$(5): Let $\curl{V_i}_{i\in I},\curl{W_i}_{i\in I}\suf \cali{P}(X)$ be locally finite refinements of $\cali{U}$ consisting of precompact open sets such that for each $i\in I,\cl{W_i}\suf V_{i}$. By Urysohn’s Lemma, there exists $\curl{f_i}_{i\in I}\in C_{c}(X;[0,1])$ such that for each $i\in I,f_{i}|_{\cl{W_i}}=1$ and $\supp{f_i}\suf V_{i}$. Let $F=\sums{}{i\in I}f_{i}$. For each $x\in X$, there exists $N_{x}\in\cali{N}^{o}(x)$ such that $\curl{i\in I:N_x\cap V_i\neq\emp}$ is finite thus

so by Gluing Lemma for Continuous Functions, $F\in C(X;\R)$. Since $\curl{W_i}_{i\in I}$ is an open cover of $X$, $F(x)>0$ for all $x\in X$. For each $i\in I$, let $g_{i}=f_{i}/F$, then $g_{i}\in C_{c}(X;[0,1])$ with $\supp{g_i}=\supp{f_i}\suf W_{i}$. For each $x\in X$, there exists $N_{x}\in\cali{N}^{o}(x)$ such that $\curl{i\in I:N_x\cap W_i\neq\emp}$ is finite thus $\curl{i\in I:g_i|_{N_x>0}}$ is also finite. Thus $\curl{g_i}_{i\in I}$ is a $C_{c}$ partition of unity subordinate to $\cali{U}$. (5)$\implies$(1): Let $\cali{U}$ be an open cover of $X$ and $\curl{f_i}_{i\in I}\suf C_{c}(X;[0,1])$ subordinate to $\cali{U}$. For each $i\in I$, le t$V_{i}=\curl{f_i>0}$, then $\curl{V_i}_{i\in I}$ is a locally finite refinement of $\cali{U}$. (5)$\implies$(6): Let $\cali{U}=\curl{X}$ (6)$\implies$(2): Let $\curl{f_i }_{i\in I}\suf C_{c}(X;[0,1])$ be a partition of unity. For each $i\in I$, let $V_{i}=\curl{f_i>0}$, then $\curl{V_i}_{i\in I}$ is a locally finite precompact open cover of $\cali{U}$.$\square$

Proof. By Proposition 3.13.7 and Lemma 3.13.3, there exists an exhaustion $\curl{U_n}_{1}^{\infty}\suf\cali{P}(X)$ of $X$ by precompact open sets. Let $U_{0}=\emp$ and $V_{n}=U_{n+1}\del \cl{U_{n-1}}$ for each $n\in\N^{+}$. For any $x\in X$, choose $n$ such that $x\in U_{n}\del U_{n-1}$. If $n>1$ then $x\in U_{n}\del\cl{U_{n-2}}=V_{n-1}$. If $n=1$ then $x\in U_{2}=V_{1}$ so $\curl{V_n}_{1}^{\infty}$ is an oepn cover of $X$. In addition, for any $m,n\in\N^{+}$, $V_{m}\cap V_{n}\neq\emp\implies \abs{n-m}\leq 1$ so $\curl{V_n}_{1}^{\infty}$ is locally finite. By Theorem 3.13.10, $X$ is paracompact.$\square$

Proof. Let $A,B\suf X$ be disjoint closed sets. By Theorem 3.13.10, there exists a partition of unity $\curl{f_i}_{i\in I}$ subordinate to $\curl{A^c,B^c}$. Let $I=I_{A}\sqcup I_{B}$ such that for each $i\in I_{A},\supp{f_i}\suf B^{c}$ and for each $i\in I_{B},\supp{f_i}\suf A^{c}$. Let $f=\sums{}{i\in I_A}f_{i}$ and $g=\sums{}{i\in I_B}f_{i}$ then $f|_{B}=0,g|_{A}=0$ hence $f|_{A}=1$. By continuity of $f$, $\curl{f\geq 2/3}\in \cali{N}_{X}(A)$ and $\curl{f\leq 1/3}\in\cali{N}_{X}(B)$ are disjoint as desired.$\square$