Subspace of Subspace is Subspace

Theorem
Let $T = \struct{S, \tau}$ be a topological space.

Let $H \subseteq S$ and $\tau_H$ be the subspace topology on $H$.

Let $K\subseteq H$.

Then the subspace topology on $K$ induced by $\tau$ equals the subspace topology on $K$ induced by $\tau_H$.

Proof
Let $\tau_K$ be the subspace topology on $K$ induced by $\tau$.

Let $\tau’_K$ be the subspace topology on $K$ induced by $\tau_H$.

Then