---- > [!definition] Definition. ([[locally finite]]) > Let $X$ be a [[topological space]]. A collection $\mathscr{A}$ of subsets of $X$ is said to be **locally finite** if each point of $X$ has a [[neighborhood]] that nontrivially intersects at most finitely many of the sets in $\mathscr{A}$. ^definition > [!basicexample] > - Any finite collection of subsets of a [[topological space]] is locally finite. > - The collection of all subsets of $\mathbb{R}$ of the form $(n,n+2)$ for $n \in \mathbb{Z}$ is locally finite. > - The collection of all subsets of the form $(-n, n)$ for natural numbers $n \in \mathbb{N}$ is *not* locally finite. ^basic-example ---- #### ---- #### 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 > ```