----- > [!proposition] Proposition. ([[T1 axiom characterization of limit points]]) > Let $X$ be a [[topological space]] satisfying the [[T1 Axiom]]; let $A \subset X$. Then the point $x$ is a [[limit point]] of $A$ if and only if every [[neighborhood]] of $x$ contains infinitely many points of $A$. > [!proof]- Proof. ([[T1 axiom characterization of limit points]]) > > If every [[neighborhood]] of $x$ intersects $A$ in infinitely many points, it certainly intersects $A$ at a point other than $x$ itself— hence is a [[limit point]]. Conversely, suppose $x$ is a [[limit point]] of $A$ and that some [[neighborhood]] $U \ni x$ intersects $A$ in only finitely many points. Then $U$ also intersects $A- \{ x \}$ in only finitely many points, $\{ x_{1},\dots,x_{n} \}=U \cap (A - \{ x \})$. By the [[T1 Axiom]], $X - \{ x_{1},\dots,x_{n} \}$ is [[open set|open in]] $X$. Then $U \cap (X - \{ x_{1},\dots,x_{n} \})$ > is [[neighborhood]] of $x$ that only trivially intersects the set $A - \{ x \}$... contradicting the assumption that $x$ is a [[limit point]] of $A$. ----- #### ---- #### 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 > ```