Definition:Hereditarily Compact Space

Definition

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

Definition 1

$T$ is hereditarily compact if and only if every subspace of $T$ is compact.

Definition 2

$T$ is hereditarily compact if and only if:

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:

$\ds \bigcup_{j \mathop \in J} U_j = \bigcup_{i \mathop \in I} U_i$

Also see

• Equivalence of Definitions of Hereditarily Compact

• Results about hereditarily compact spaces can be found here.