Thickenings, Neighbourhoods with Compact Closure, Urysohn Functions and Partitions of Unity in a Locally Compact Metric Space

In the setting of Locally Compact Metric Spaces and Compactly Supported Functions: Standing Notation, let $(X,d)$ be a locally compact metric space and let $K\subseteq X$ be compact. Let $V\subseteq X$ be open with $K\subseteq V$.

(i) Compact thickening. If $K\neq\emptyset$, there is a real number $\delta>0$ such that the set

$$\{x\in X:\operatorname{dist}(x,K)\le\delta\}$$

is compact and contained in $V$.

(ii) Compact neighbourhood. There is an open set $W\subseteq X$ with $K\subseteq W$, $\operatorname{cl}W$ compact and $\operatorname{cl}W\subseteq V$.

(iii) Urysohn function. There is a function $f$ in $C_{c}(X)$ with $0\le f(x)\le1$ for every $x\in X$, $f(x)=1$ for every $x\in K$, and support $\operatorname{supp}f\subseteq V$.

(iv) Partition of unity. Let $n\in\mathbb{N}$ and let $V_{1},\dots,V_{n}\subseteq X$ be open with $K\subseteq V_{1}\cup\dots\cup V_{n}$. Then there are $h_{1},\dots,h_{n}\in C_{c}(X)$ with $0\le h_{i}(x)\le1$ for every $x\in X$ and $\operatorname{supp}h_{i}\subseteq V_{i}$, for every $i\in[n]$, and

$$\sum_{i=1}^{n}h_{i}(x)\le1\quad(x\in X),\qquad\sum_{i=1}^{n}h_{i}(x)=1\quad(x\in K).$$