Pathwise-Connected

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