Closed Subsets of Compact Sets Are Compact, and Compact Subsets of a Metric Space Are Closed and Bounded

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces:

¶ Let $(X,\mathcal{T})$ be a topological space, let $K\subseteq X$ be compact, and let $C\subseteq K$ be closed in $(X,\mathcal{T})$. Then $C$ is compact in $(X,\mathcal{T})$.

¶ Let $(X,d)$ be a metric space, and let $K\subseteq X$ be compact in $(X,\mathcal{T}_{d})$. Then $K$ is closed in $(X,\mathcal{T}_{d})$ and bounded in $(X,d)$.