Definition 2

A subset A of a topological space X is said to be a quasi-compact (resp. compact) set if the subspace A is quasi-compact (resp. compact).

A subset $A$ of a topological space $X$ is a quasi-compact set if and only if every covering of $A$ by open sets of $X$ contains a finite covering of $A$; this follows from axiom $(C''')$. In a Hausdorff space, the notions of quasi-compact and compact sets are the same, since every subspace is Hausdorff.