Definition:Compact Space/Topology/Subspace

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $H \subseteq S$ be a subset of $S$.

Also known as
A subset $H$ of $S$ such that $\left({H, \tau_H}\right)$ is a compact subspace of $T$ is often referred to as a compact set or compact subset of $T$.

This can be seen as another instance of the ubiquity of identifying a topological space with its underlying set.

Also see

 * Equivalence of Definitions of Compact Topological Subspace


 * Definition:Relatively Compact