Nets and compactness in topological spaces.

See Theorem $15.3$ in this excellent PDF, Translating Between Nets and Filters, by Saitulaa Naranong; it’s well worth reading the whole thing.

Suppose that $x_\alpha$ is a net in $X$ with no convergent subnet. Then $(x_\alpha)$ do not have accumulation point in $X$. For each $x\in X$, let $V_x$ be an open neighbourhood of $x$ that excludes all the part of the net from some term onward. Let $V=\{V_x:\ x\in X\}$ and note that $V$ is an open cover of $X$. Can you prove that it is impossible to find a finite subcover of $X$ in $V$?

On the other hand, let $V$ be a open cover of $X$ such that every finite subcover of $V$ do not cover $X$. Consider the open cover $U$ of $X$ consisting of finite unions of elements of $V$. If $A,B\in U$, we say that $A\leq B$ when $A\subseteq B$. With this relation, $V$ is an directed set. For $A\in V$, let $x_A\in X\setminus A$. Can you show that the net $x_A$ does not have any convergent subnet?