Definition:Compact Space/Topology/Subspace

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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.