Definition:Compact Space/Topology/Subspace

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

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

Equivalence of Definitions
The proof that these definitions are equivalent is given in Equivalent Definitions of Compact Topological Subspace.

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 $T$.

Note, however, that the notion of a compact subspace requires the subspace topology in order to be defined.

Thus, whenever the terms compact set or compact subset are used, it is important to remember that what is really meant is the term compact subspace.