Subset of a Precompact Set is Precompact

Last updated Nov 1, 2022

Let $X$ be a Topological Space and let $S \subset X$ be Precompact. Then if $R \subset S$, $R$ is Precompact as well.

Because $\text{cl}S \supset R$, so $\text{cl}S \supset \text{cl}R$ by definition of Closure. Since $\text{cl}S$ is Compact and Closed Subset of a Compact Set is Compact, $\text{cl}R$ is Compact. Thus $R$ is Precompact. $\blacksquare$