\begin{definition}[Definition:Hereditarily Compact Space/Definition 2]
Let $T = \struct {S, \tau}$ be a topological space.
$T$ is ''' hereditarily compact''' {{iff}}:
:for each family $\family {U_i}_{i \mathop \in I}$ of open sets of $T$, there exists a finite subset $J \subset I$ such that:
:::$\ds \bigcup_{j \mathop \in J} U_j = \bigcup_{i \mathop \in I} U_i$
\end{definition}