Let $X$ be a metric space, $K\subseteq X$ be compact and $C\subseteq X$ be closed. Use the definition to show that $K\cap C$ is compact.
I am unsure as to which "definition" the question refers to. I think it has to do with the Heine-Borel Theorem [A subset of $\mathbb R^m$ is compact if and only if it is closed and bounded] but I am not sure. I have attempted it by using the fact that $C$ is closed (and therefore bounded), so $K\cap C$ must also be closed (and bounded), but I don't know where to go after this or how to show it topologically
This is what you have to do.
Let $A_\alpha$ be an open cover of $C$. Adjoin $C^c$ to this open cover, where $C^c$ denotes the complement of $C$.
We now have an open cover of $K$. However, since $K$ is compact, then there is a finite cover of $K$.
Remove $C^c$ from this finite cover if it is in it, and then you have a finite covering from $A_\alpha$.