Compact Topological Space and Compact Subset

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

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

$$X\subseteq\bigcup_{i\in I}U_i,$$

there exists a finite subset $J\subseteq I$ such that

$$X\subseteq\bigcup_{i\in J}U_i,$$

where a union indexed by the empty set is empty.

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.