# Definition:Compact Space/Topology/Subspace

## Definition

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

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

### Definition 1

The topological subspace $T_H = \left({H, \tau_H}\right)$ is **compact in $T$** if and only if $T_H$ is itself a compact topological space.

### Definition 2

$H$ is **compact in $T$** if and only if every open cover $\mathcal C \subseteq \tau$ for $H$ has a finite subcover.

## 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 terminology is appropriate when the subspace topology $\tau_H$ is not directly relevant, and so its construction does not need to be considered.

However, it is worth understanding that a **compact (sub)set** *becomes* a **compact space** by the operation of applying that subspace topology.

Hence it is important to bear in mind that the **compactness** of a set has meaning *only* in the context of the topological space into which it is embedded.

## Also see

- Results about
**compact spaces**can be found here.