on the spaces and For example, if is a Hausdorff space then must be a closed set. Conversely, if is a Hausdorff space and is a closed set, then the coimage of if given the quotient space topology, must also be a Hausdorff space. A space is compact if and only if the kernel of every family of closed subsets having the finite intersection property (FIP) is non-empty; said differently, a space is compact if and only if every family of closed subsets with F.I.P. is fixed. See also References Bibliography Abstract algebra Basic concepts in set theory Set theory Topology