Intersection of Closed Set with Compact Subspace is Compact/Proof 1

Proof
Let topological space $T_K = \left({K, \tau_K}\right)$ be a (topological) subspace of $T$.

By Closed Set in Topological Subspace, $H\cap K$ is closed in $T_K$.

By Closed Subspace of Compact Space is Compact, $H\cap K$ is compact in $T_K$.

Therefore

So, $H\cap K$ is compact in $T$.