----- > [!proposition] Proposition. ([[connected components versus path-connected components]]) > Let $X$ be a [[topological space]]. Each [[path-connected component]] of $X$ lies in a [[connected component]] of $X$. If $X$ is [[locally connected, locally path-connected|locally path-connected]], then the [[connected component]]s and the [[path-connected component]]s of $X$ are in fact the same. > [!proof]- Proof. ([[connected components versus path-connected components]]) >That each path-connected component lies in exactly one connected component of $X$ is immediate from the justification in [[connected component]], since path-connected components are connected subsets. > Suppose $X$ is [[locally connected, locally path-connected|locally path-connected]] and let $C$ be a [[connected component]] of $X$. We want to show that for all $x \in C$, $C$ is also the [[path-connected component]] $P$ of $x$. Since $P$ is [[connected]], $P \subset C$. Suppose the inclusion is proper: $P \subsetneq C$. Let $Q$ be the union of all the [[path-connected component]]s of $X$ other than $P$ which nontrivially intersect $C$. Note that each must lie in $C$, so that $C=P \cup Q$. The result [[locally connected iff open sets have open components]] implies $P$ and $Q$ are each open, yielding a [[separation of a topological space|separation]] of $C$: contradiction. ----- #### ---- #### 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 > ```