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 exist a 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,

$$A\subseteq U_{i_1}\cup\cdots\cup U_{i_n}.$$