Definition: Let $(X,d)$ be a metric space. We say $K \subseteq X$ is a compact subset of $X$ if every open cover for $K$ has a finite sub-cover.
Fact: $(K,d')$ can be regarded as a metric space where $d'$ is restriction of $d$.
Question: If $K$ is a compact subset of $(X,d)$. Is $(K,d')$ a compact space?
The subspace topology for any compact subset of a space is compact.
Likewise, if a space is compact and is mapped continuously into another space, including if it is embedded as a subspace, then that image/embedding is a compact subset of the space.