Indiscrete Space is Hereditarily Compact

Theorem
Let $\left({S, \tau}\right)$ be an indiscrete topological space.

Then $\left({S, \tau}\right)$ is hereditarily compact.

Proof
Let $\left({H, \tau_H}\right)$ be a subspace of $T$.

From Subset of Indiscrete Space is Compact and Sequentially Compact, $\left({H, \tau_H}\right)$ is compact.

The result follows by definition of hereditarily compact.

Also see

 * Hausdorff Space is Hereditarily Compact iff Finite