---- > [!definition] Definition. ([[compact embedding of topological subspaces]]) > Let $X$ be a [[topological space]] and let $V,W$ be subsets of $X$. We say $V$ is **compactly embedded in $W$**, and write $V \Subset W$, if > 1. $\overline{V}$ is [[compact]] (i.e. $V$ is [[precompact]]); > 2. $\overline{V} \subset \text{int }W$. > > Equivalently for $X$ [[Hausdorff space|Hausdorff]]: there exists some [[compact]] set $K$ for which $V \subset K \subset \text{int }W$.[^1] > > Here, $\overline{V}$ denotes the [[closure]] of $V$ in $X$ and $\text{int }W$ denotes the interior of $W$. [^1]: Assume there exists [[compact]] $K$ for which $V \subset K \subset \text{int }W$. Then $\overline{V} \subset K$ since $K = \overline{K}$ by [[Hausdorff space|Hausdorffness]] of $X$. [[Closed subspaces of compact spaces are compact]], $\overline{V}$ is [[compact]]. Conversely, if $(1)$ and $(2)$ hold then put $K=\overline{V}$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```