Definition:Hereditarily Compact Space/Definition 2

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

$T$ is  hereditarily compact :
 * for each family $\left\langle{U_i}\right\rangle_{i \mathop \in I}$ of open sets of $T$, there exists a finite subset $J \subset I$ such that:
 * $\displaystyle \bigcup_{j \mathop \in J} U_j = \bigcup_{i \mathop \in I} U_i$

Also see

 * Equivalence of Definitions of Hereditarily Compact