----- > [!proposition] Proposition. ([[local compactness condition for subspaces of locally compact Hausdorff spaces]]) > Let $X$ be a [[locally compact]] [[Hausdorff space]]. Then > - Open [[subspace topology|subspaces]] of $X$ are [[locally compact]]. > - [[closed set|Closed]] [[subspace topology|subspaces]] of $X$ are [[locally compact]]. > - The intersection of an open and a closed set in $X$ is [[locally compact]]. > [!proof]- Proof. ([[local compactness condition for subspaces of locally compact Hausdorff spaces]]) > Suppose that $A$ is closed in $X$. Given $x \in A$, let $C$ be a [[compact]] [[subspace topology|subspace]] of $X$ containing the [[neighborhood]] $U$ of $x$ in $X$. Then $C \cap A$ is closed in $C$ and thus [[compact]], and it contains the [[neighborhood]] $U \cap A$ of $x$ in $A$. (We have not used the Hausdorff condition here.) > Suppose now that $A$ is open in $X$. Given $x \in A$, we apply the [[local compactness characterization for Hausdorff spaces]] to choose a [[neighborhood]] $V$ of $x$ in $X$ such that $\overline{V}$ is [[compact]] and $\overline{V} \subset A$. Then $C=\overline{V}$ is a [[compact]] [[subspace topology|subspace]] of $A$ containing the [[neighborhood]] $V$ of $x$ in $A$. > Suppose $A$ is open in $X$ and $B$ is closed in $X$. Given $x \in A \cap B$, we apply the [[local compactness characterization for Hausdorff spaces]] to choose a [[neighborhood]] $V$ of $x$ in $X$ such that $\overline{V}$ is [[compact]] and $\overline{V} \subset A \cap B$. Then $C=\overline{V}$ is a [[compact]] [[subspace topology|subspace]] of $A\cap B$ containing the [[neighborhood]] $V$ of $x$ in $A \cap B$. ^ee0102 ----- #### ---- #### 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 > ```