Pathwise-Connected

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.