Compactness of Intersections with Closed Sets and of Level Sets of Semicontinuous Functions

Let $(X,d)$ be a metric space, equipped with the topology $\mathcal{T}_d$ of its metric-open subsets, a topology by Metric Open Sets Form a Topology. Let $\mathbb{R}$ be the ordered field of real numbers and let $K\subseteq X$ be compact in $X$. Then the following hold.

1. (Intersection with a closed set) If $C\subseteq X$ is closed in $(X,\mathcal{T}_d)$, then $K\cap C$ is compact in $X$.

2. (Superlevel sets) If $w:K\to\mathbb{R}$ is upper semicontinuous on $K$ and $c\in\mathbb{R}$, then the set

$$\{x\in K:\ c\le w(x)\}$$

is compact in $X$.

3. (Sublevel sets) If $w:K\to\mathbb{R}$ is lower semicontinuous on $K$ and $c\in\mathbb{R}$, then the set

$$\{x\in K:\ w(x)\le c\}$$

is compact in $X$.