compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has...

topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely...

X} . Every metric space is naturally a topological space, and for metric spaces, the notions of compactness and sequential compactness are equivalent (if...

space is said to be σ-compact if it is the union of countably many compact subspaces. A space is said to be σ-locally compact if it is both σ-compact...

a topological space X {\displaystyle X} is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made...

example of a countably compact space that is not compact. Every compact space is countably compact. A countably compact space is compact if and only if it...

ambient space). The term precompact (or pre-compact) is sometimes used with the same meaning, but precompact is also used to mean relatively compact. These...

of a compact topological space is relatively compact (since a closed subset of a compact space is compact). And in an arbitrary topological space every...

by Dieudonné (1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only...

continuous. Functions with compact support on a topological space X {\displaystyle X} are those whose closed support is a compact subset of X . {\displaystyle...