---- > [!definition] Definition. ([[path-connected]]) > A [[topological space]] $X$ is said to be **path-connected** if every pair of points in $X$ be joined by a [[parameterized curve]] in $X$, i.e., the [[path-connected component|0th homotopy group]] $\pi_{0}(X)$ is trivial. > \ > [[path-connected versus connected|Note the relationship to]] [[connected]] spaces. > [!basicexample] > The unit ball $B^{n} \subset \mathbb{R}^{n}$, $B^{n}=\{ x : \|x\|_{2} \leq 1 \},$ where $\|\cdot\|_{2}$ denotes the [[Lp-norm|euclidean norm]], is a [[path-connected]] subset of $\mathbb{R}^{n}$. Indeed, it is [[convex]]: [[euclidean balls are convex]] (the paths being just [[line]]s). ---- #### ---- #### 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 > ```