---- > [!definition] Definition. ([[locally connected, locally path-connected]]) > A [[topological space]] $X$ is said to be **locally connected at $x \in X$** if for every open [[neighborhood]] $U$ of $x$ there is a [[connected]] [[neighborhood]] $V$ of $x$ contained in $U$. If $X$ is locally connected at each of its points, it is said simply to be locally connected. > \ > That is, $X$ is **locally connected** if its open connected sets [[nestles in|nestle in]] its open sets. > \ > Similarly, $X$ is said to be **locally path-connected at $x$** if for every [[neighborhood]] $U$ of $x$, there is a [[path-connected]] [[neighborhood]] $V$ of $x$ contained in $U$. If $X$ is locally path-connected at each of its points, then it is said to be locally path-connected. > [!basicexample] >- Each interval and each [[ray]] in the real line is *both* [[connected]] and locally connected >- The [[subspace topology|subspace]] $[-1,0) \cup (0,1]$ of $\mathbb{R}$ is not connected, but is locally-connected > [!basicnonexample] > - The [[topologist's sine curve]] is [[connected]] but not locally connected > - The rationals $\mathbb{Q} \subset \mathbb{R}$ are neither connected nor locally connected. > [!equivalence] > - [[locally connected iff open sets have open components]] ---- #### ---- #### 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 > ```