Compact Topological Space and Compact Subset

Let $(X,\mathcal{T})$ be a topological space.

We say that $X$ is compact if for every set $A$ and every family of subsets of $X$ $(U_a)_{a\in A}$ such that $U_a\in\mathcal{T}$ for every $a\in A$ and

there exist a natural number $n\in\mathbb{N}$ and elements $a_1,\dots,a_n\in A$ such that

If $A\subseteq X$, we say that $A$ is compact in $X$ if $A$ is compact as a topological space equipped with the subspace topology.