If $J$ is a directed set and $X$ is a topological space, then a net in $X$ is a function $f\colon J \to X$. A net converges to $x \in X$ if for every neighborhood $U$ of $x$ there is $\alpha \in J$ so that $\alpha \preceq \beta \implies f(\beta)\in U$.
Given a net $f \colon J \to X$, a subnet is a net $f\circ g \colon K \to X$ where $g \colon K \to J$ is a map of directed sets so that $g(K)$ is cofinal in $J$.
Theorem. $X$ is compact iff every net in $X$ has a convergent subnet.
So, rather than probing your space with sequences to determine compactness, you instead probe your space with directed sets.