Definition 3

A subset $A$ of a topological space $X$ is said to be relatively quasi-compact (resp. relatively compact) in $X$ if $A$ is contained in a quasi-compact (resp. compact) subset of $X$.

To abbreviate, we say also that $A$ is a "relatively quasi-compact set" (resp. "relatively compact set") when there is no ambiguity about $X$. In a Hausdorff space the notions of relatively quasi-compact set and relatively compact set are the same.