Indiscrete Space is Hereditarily Compact

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.

Also see

 * Hausdorff Space is Hereditarily Compact iff Finite