Let $(X,\mathcal{T})$ be a \reftext{def:topological-space-2026a}{topological space}, and let $A\subseteq X$. Then the following are equivalent.

1. The subset $A$ is \reftext{def:compact-space-and-subset-2026a}{compact in $X$}.
2. For every \reftext{def:open-cover-subcover-topological-space-2026a}{open cover} $(U_i)_{i\in I}$ of $A$ in $X$, there exist a \reftext{def:natural-numbers-2026a}{natural number} $n\in\mathbb{N}$ and elements $i_1,\dots,i_n\in I$ such that $(U_{i_1},\dots,U_{i_n})$ is a finite subcover of $A$, that is,