Proof. (1) $\Rightarrow$ (2): For each $x \in X$, there exists a precompact open neighbourhood $U_{x} \in \cn^{o}(x)$. Since $\bracs{U_x| x \in X}$ is an open cover of $X$, there exists a locally finite refinement $\mathcal{V}$. For each $V \in \mathcal{V}$, there exists $x \in X$ such that $V \subset U_{x}$. In which case, $\ol{V}\subset \ol{U_x}$ is compact.

(2) $\Rightarrow$ (3): Let $\cf \subset 2^{X}$ be a locally finite open cover of $X$ consisting of precompact open sets. By Lemma 4.20.7, there exists a locally finite open cover $\bracs{G_F}_{F \in \cf}$ of $X$ consisting of precompact open sets such that $\ol{F}\subset G_{F}$ for all $F \in \cf$.

For each $F \in \cf$, let

then $\mathcal{U}_{F}$ is a precompact open cover of $\ol{F}$. By compactness of $\ol{F}$, there exists $\mathcal{V}_{F} \subset \mathcal{U}_{F}$ finite such that $\ol{F}\subset \bigcup_{V \in \mathcal{V}_F}V$.

Let $\mathcal{V}= \bigcup_{F \in \cf}\mathcal{V}_{F}$, then $\mathcal{V}$ is a precompact open cover of $X$. For any $x \in X$, there exists $N \in \cn(x)$ such that $\bracs{F \in \cf|N \cap G_F}$ is finite. Thus

is finite, and $\mathcal{V}$ is locally finite.

(3) $\Rightarrow$ (4): By Lemma 4.20.7.

(4) $\Rightarrow$ (5): Let $\seqi{V}, \seqi{W}\subset 2^{X}$ be locally finite refinements of $\mathcal{U}$ consisting of precompact open sets such that for each $i \in I$, $\ol{W_i}\subset V_{i}$.

By Urysohn’s lemma, there exists $\seqi{f}\in C_{c}(X; [0, 1])$ such that for each $i \in I$, $f_{i}|_{\ol{W_i}}= 1$ and $\supp{f_i}\subset V_{i}$.

Let $F = \sum_{i \in I}f_{i}$. For each $x \in X$, there exists $N_{x} \in \cn^{o}(x)$ such that $\bracs{i \in I|N_x \cap V_i \ne \emptyset}$ is finite. In which case,

thus $F|_{N_x}\in C(N_{x}; \real)$. By Lemma 4.6.2, $F \in C(X; \real)$.

Since $\seqi{W}$ 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 $\supp{g_i}= \supp{f_i}\subset W_{i}$. For any $x \in X$, there exists $N_{x} \in \cn^{o}(x)$ such that $\bracs{i \in I|N_x \cap W_i \ne \emptyset}$ is finite. In which case, $\bracs{i \in I|0 < g_i|_{N_x}}$ is also finite. Thus $\seqi{g}$ is a $C_{c}$ partition of unity subordinate to $\mathcal{U}$.

(5) $\Rightarrow$ (1): Let $\mathcal{U}$ be an open cover of $X$ and $\seqi{f}\subset C_{c}(X; [0, 1])$ subordinate to $\mathcal{U}$. For each $i \in I$, let $V_{i} = \bracs{f_i > 0}$, then $\seqi{V}$ is a locally finite refinement of $\mathcal{U}$.

(5) $\Rightarrow$ (6): Take $\mathcal{U}= \bracs{X}$.

(6) $\Rightarrow$ (2): Let $\seqi{f}\subset C_{c}(X; [0, 1])$ be a partition of unity. For each $i \in I$, let $V_{i} = \bracs{f_i > 0}$, then $\seqi{V}$ is a locally finite precompact open cover of $\mathcal{U}$.$\square$