Tag Info

The compactness tag is for questions about compactness and its many variants (e.g. sequential compactness, countable compactness) as well locally compact spaces; compactifications (e.g. one-point, Stone-Čech) and other topics closely related to compactness. This includes logical compactness.

Compactness is a topological property. We say that a topological space $$X$$ is compact if whenever we cover $$X$$ by a collection of open sets we can find a finite number of open sets from the collection which cover $$X$$. For example, $$[0,1]$$ is a compact subspace of $$\mathbb{R}$$, but $$(0,1)$$ and $$\mathbb{R}$$ are not.

We say that a space $$X$$ is sequentially compact if every sequence has a convergent subsequence. These properties are equivalent for metric space, although neither implies the other in general.

This tag may also be used for questions about logical compactness, such as the compactness theorem.