Compactness is a topological property. We say that a topological space $$X$$ is compact if whenever we cover $$X$$ by a collection of open sets we can find a finite number of open sets from the collection which cover $$X$$. For example, $$[0,1]$$ is a compact subspace of $$\mathbb{R}$$, but $$(0,1)$$ and $$\mathbb{R}$$ are not.
We say that a space $$X$$ is sequentially compact if every sequence has a convergent subsequence. These properties are equivalent for metric space, although neither implies the other in general.