A topological space X is pathwise-connected iff for every two points x,y in X, there is a continuous function f from [0,1] to X such that f(0)=x and f(1)=y. Roughly speaking, a space X is pathwise-connected if, for every two points in X, 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 also

Connected, Connected Space, Convex, CW-Complex, Locally Pathwise-Connected, Star Convex, Topological Space

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Cite this as:

Weisstein, Eric W. "Pathwise-Connected." From MathWorld--A Wolfram Web Resource.

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