----- > [!proposition] Proposition. ([[local compactness characterization for Hausdorff spaces]]) > Let $X$ be a [[Hausdorff space]]. The following are equivalent: >- $X$ is [[locally compact]]; >- Each point of $X$ has a [[precompact]] [[neighborhood]]; >- $X$ has a [[basis for a topology|basis]] of [[precompact]] open subsets >- Perhaps more strongly, given $x \in X$, and given an open [[neighborhood]] $U \ni x$, there is an [[neighborhood|open neighborhood]] $V \ni x$ whose [[closure]] $\overline{V}$ is [[compact]] (i.e. $V$ is [[precompact]]) with $\overline{V} \subset U$. > [!proof]- Proof. ([[local compactness characterization for Hausdorff spaces]]) > Clearly $(3) \implies (2) \implies (1)$. We'll show $(1) \implies (3)$ > Suppose $X$ is [[locally compact]], let $x \in X$. It will suffice to obtain a [[local basis]] of [[precompact]] open subsets. Obtain a [[compact]] set $K$ containing $x$ and an open set $U$ satisfying $x \in U \subset K$. The collection $\mathcal{V}$ of all neighborhoods of $x$ contained in $U$ is clearly a local basis at $x$. If $V \subset U$, then $\overline{V} \subset K$ because $K$ is [[closed set|closed]] in $X$ ([[every compact subspace of a Hausdorff space is closed]]). Therefore $\overline{V}$ is [[compact]], by [[closed subspaces of compact spaces are compact]]. Hence $V$ is [[precompact]]. Thus $\mathcal{V}$ is the required local basis. ----- #### ---- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM outgoing([[]]) FLATTEN file.tags GROUP BY file.tags as Tag