Compact Subsets of a Metric Space: Covers by Open Sets, Finite Unions and Increasing Covers; Sublevel Sets of the Distance and Closed Balls

In the setting of Locally Compact Metric Spaces and Compactly Supported Functions: Standing Notation, let $(X,d)$ be a metric space.

(i) Covers by open sets. A subset $K\subseteq X$ is compact if and only if for every set $\mathcal{U}$ of open subsets of $X$ with $K\subseteq\bigcup\mathcal{U}$ there is a finite subset $\mathcal{V}\subseteq\mathcal{U}$ with $K\subseteq\bigcup\mathcal{V}$.

(ii) Finite unions. If $\mathcal{K}$ is a finite set of compact subsets of $X$, then $\bigcup\mathcal{K}$ is compact.

(iii) Increasing covers. Let $K\subseteq X$ be compact and let $(U_{m})_{m\in\mathbb{N}}$ be a family of open subsets of $X$ with $U_{m}\subseteq U_{m+1}$ for every $m\in\mathbb{N}$ and $K\subseteq\bigcup_{m\in\mathbb{N}}U_{m}$. Then $K\subseteq U_{M}$ for some $M\in\mathbb{N}$.

(iv) Closed balls. Every closed ball $\bar{B}(x,r)$, with $x\in X$ and real $r\ge0$, is closed.

Let $A\subseteq X$ be nonempty and let $\delta$ be a real number.

(v) Open sublevel sets of the distance. The set $\{x\in X:\operatorname{dist}(x,A)<\delta\}$ is open.

(vi) Closed sublevel sets of the distance. The set $\{x\in X:\operatorname{dist}(x,A)\le\delta\}$ is closed.