Indiscrete Space is Hereditarily Compact
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Theorem
Let $\struct {S, \tau}$ be an indiscrete topological space.
Then $\struct {S, \tau}$ is hereditarily compact.
Proof
Let $\struct {H, \tau_H}$ be a subspace of $T$.
From Subset of Indiscrete Space is Compact, $\struct {H, \tau_H}$ is compact.
The result follows by definition of hereditarily compact.
$\blacksquare$