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 $\left\langle{U_i}\right\rangle_{i \mathop \in I}$ 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

  • Results about hereditarily compact spaces can be found here.