What is Incompactness in Topology?

Incompact is a term used in topology and related fields to describe a space that is not compact. A compact space is one that is "finished" or "closed off" in some sense, meaning that it does not contain any "holes" or "gaps" that can be filled by adding points or open sets.

In contrast, an incompact space is one that has holes or gaps that cannot be filled by adding points or open sets. For example, the set of all real numbers is incompact because it has no endpoints and there are no open sets that can be added to it to make it compact.

Incompactness is often used as a property of spaces that are not intended to be compact or complete in some sense. For example, the set of all real numbers is incompact because it is not intended to be a complete space, but rather a space that contains all the real numbers in a certain sense.