Every subset of a compact topological space is relatively compact (since a closed subset of a compact space is compact). In an arbitrary topological space every subset of a relatively compact set is relatively compact. Every compact subset of a
Hausdorff space is relatively compact. In a non-Hausdorff space, such as the
particular point topology on an infinite set, the closure of a compact subset is
not necessarily compact; said differently, a compact subset of a non-Hausdorff space is not necessarily relatively compact. Every compact subset of a (possibly non-Hausdorff)
topological vector space is
complete and relatively compact. In the case of a
metric topology, or more generally when
sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence in has a subsequence convergent in . Some major theorems characterize relatively compact subsets, in particular in
function spaces. An example is the
Arzelà–Ascoli theorem. Other cases of interest relate to
uniform integrability, and the concept of
normal family in
complex analysis.
Mahler's compactness theorem in the
geometry of numbers characterizes relatively compact subsets in certain non-compact
homogeneous spaces (specifically spaces of
lattices). ==Counterexample==