Compactness

A compact Topological Space is a topological space such that every open cover has a finite subcover. That is, if $\mathbb{U}$ is a collection of open sets $U$ that cover $X$, then there exists a subset $V$ of $\mathbb{U}$ such that $V$ is finite and covers $X$.

An equivalent definition is that of in terms of nets; a set is compact if and only if all universal nets converge. We will prove this in this article, as well as several basic properties and definitions related to compactness.