If I have some subset of a metric space, is it always possible to determine if it is compact? If so, how?
It seems to be quite easy to show something is not compact(in terms of what is required: a single counter example), but I can't see how you would determine if it is compact(having no counter example).
What I know:
A subset $K$ of a metric space $X$ is said to be compact if every open cover of $K$ contains a finite subcover.
Well, you could use the Heine Borel theorem here. It roughly says, that for a finite dimensional normed vector space, with the topology provided by the norm, a subset is compact, if and only if it is closed and bounded. Metric spaces which have this property are said to have the Heine Borel property. However, the Heine–Borel theorem can be generalized to arbitrary metric spaces by strengthening the conditions required for compactness:
A subset of a metric space is compact if and only if it is complete and totally bounded.
See this Wikipedia article for further discussion, under the generalizations section. So, try to determine if you are after a general metric space with no notion of addition or scalar multiplication, then you could use the generalized Heine Borel theorem.