Compact Subset Criterion via Open Covers in the Ambient Space

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

1. The subset $A$ is compact in $X$.
2. For every open cover $(U_i)_{i\in I}$ of $A$ in $X$ there exists a finite subset $J\subseteq I$ such that the subfamily $(U_j)_{j\in J}$ is a subcover of $A$, that is,

$$A\subseteq\bigcup_{j\in J}U_j,$$

where a union indexed by the empty set is empty.