----- > [!proposition] Proposition. ([[locally connected iff open sets have open components]]) > A [[topological space]] $X$ is [[locally connected, locally path-connected|locally connected]] if and only if for every open set $U$ of $X$, each [[connected component]] of $U$ is open in $X$. > \ > Similarly, $X$ is locally [[path-connected]] if and only if for every open set $U$ of $X$, each [[path-connected component]] of $U$ is open in $X$. > [!proof]- Proof. ([[locally connected iff open sets have open components]]) > ~ > $\to.$ Suppose $X$ is [[locally connected, locally path-connected|locally connected]] and let $U \subset X$ be open in $X$. Let $C$ be a [[connected component]] of $U$. If $x$ is a point of $C$, by local connectedness we can choose a connected [[neighborhood]] $V$ of $x$ s.t. $V \subset U$. Since $V$ is [[connected]], it must lie entirely in the component $C$ of $U$. Therefore, $C$ is open in $X$ because every point $x \in C$ is an [[topological interior|interior point]] of $C$. > $\leftarrow.$ Suppose that for every open set of $X$ each [[connected component]] of that set is open in $X$. Given $x \in X$ and an open [[neighborhood]] $U \ni x$, let $C$ be the [[connected component]] of $U$ containing $x$. Since $C$ is open in $X$, it witnesses that $X$ is [[locally connected, locally path-connected|locally connected]]. ----- #### ---- #### 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 > ```