Definition:Hereditarily Compact Space
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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
- Results about hereditarily compact spaces can be found here.