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 $\cali{U}$.$\square$