----- > [!proposition] Proposition. ([[limits in Hausdorff spaces are unique]]) > If $X$ is a [[Hausdorff space]], then a [[sequence]] of points of $X$ [[converge]]s to at most one point of $X$. > [!proof]- Proof. ([[limits in Hausdorff spaces are unique]]) > Suppose $x_{1},x_{2},\dots$ is a [[sequence]] of points in $X$ converging to $x \in X$ and also $y \in X$. Consider an arbitrary [[neighborhood]] $U \ni x$; there exists $N \in \mathbb{N}$ s.t. for all $n \geq N$, $x_{n} \in U$. \ Since $x_{1},x_{2},\dots$ [[converge]]s also to $y \in X$, for an arbitrary [[neighborhood]] $V \ni y$, there exists $M \in \mathbb{N}$ s.t. for all $n \geq M$, $x_{n} \in V$. Set $P=\max(N,M)$. Then for all $n \geq P$ we have $x_{n} \in U$ and $x_{n} \in V$, implying that $U \cap V \neq \emptyset$. \ The [[Hausdorff space|Hausdorff Axiom]] proceeds to enforce that $x=y$. \ (Munkres is more concise, but this was nice bc it let me go through defs etc) ----- #### ---- #### 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 > ```