Definition 2.1:

A space $X$ is said to be compact if every open cover of $X$ has a finite sub-cover. If $X$ is a metric space, this is equivalent to the statement that every sequence has a convergent subsequence.

If $X$ is a topological space and $A$ is a subset of $X$ we say that $A$ is compact if it is so as a topological space with the subspace topology. This is the same as saying that every covering of $A$ by open sets in $X$ admits a finite subcovering. It is clear that a closed subset of a compact subset is necessarily compact. However a compact set need not be closed as can be seen by looking at $X$ endowed the indiscrete topology, where every subset of $X$ is compact. However, if $X$ is a Hausdorff space then compact subsets are necessarily closed. We shall always work with Hausdorff spaces in this course. For subsets of $\mathbb{R}^n$ we have the following powerful result.