Continuous Partition of Unity Subordinate to a Finite Open Cover of a Compact Set in a Metric Space

Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$, which is a topology on $X$ by Metric Open Sets Form a Topology. Let $\mathbb{R}$ denote the real numbers, with the addition, multiplication, identities, additive inverses, multiplicative inverses and order of their ordered field structure; for $s,t\in\mathbb{R}$ write $s-t$ for $s+(-t)$, write $s<t$ to mean that $s\le t$ and $s\ne t$, write $s^{-1}$ for the multiplicative inverse of $s$ when $s\ne 0$, let $|s|$ be the absolute value of $s$, and let $d_{\mathbb{R}}$ be given by $d_{\mathbb{R}}(s,t)=|s-t|$, which is a metric on $\mathbb{R}$ by The Absolute Value Metric on the Real Line. Sums of the form $\sum_{i=1}^{n}$ are the finite sums of that definition.

Let $C\subseteq X$ be compact in $(X,\mathcal{T}_d)$. Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$, and let $(U_i)_{i\in[n]}$ be a family of subsets of $X$ such that $U_i\in\mathcal{T}_d$ for every $i\in[n]$ and

$$C\subseteq\bigcup_{i\in[n]}U_i .$$

Then there exist maps $h_i:X\to\mathbb{R}$ and subsets $D_i\subseteq X$, one for each $i\in[n]$, such that the following hold.

1. (Cutoffs) For every $i\in[n]$ the map $h_i$ is continuous on $X$ as a map from $(X,d)$ to $(\mathbb{R},d_{\mathbb{R}})$, and $0\le h_i(x)\le 1$ for every $x\in X$.

2. (Subordination) For every $i\in[n]$ the set $D_i$ is closed in $(X,\mathcal{T}_d)$, $D_i\subseteq U_i$, and $h_i(x)=0$ for every $x\in X$ with $x\notin D_i$.

3. (Subpartition) $\sum_{i=1}^{n}h_i(x)\le 1$ for every $x\in X$.

4. (Partition on $C$) $\sum_{i=1}^{n}h_i(x)=1$ for every $x\in C$.