A topological space is pathwise-connected iff for every two points , there is a continuous function from [0,1] to such that and . Roughly speaking, a space is pathwise-connected if, for every two points in , there is a path connecting them. For locally pathwise-connected spaces (which include most "interesting spaces" such as manifolds and CW-complexes), being connected and being pathwise-connected are equivalent, although there are connected spaces which are not pathwise-connected. Pathwise-connected spaces are also called 0-connected.
See alsoConnected, Connected Space, Convex, CW-Complex, Locally Pathwise-Connected, Star Convex, Topological Space
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Weisstein, Eric W. "Pathwise-Connected." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Pathwise-Connected.html