A topological space
connected or some such term) if for any two points P and Q in it there is a continuous
map f from [0, 1] such that f(x) is in the space, f(0) is P, and f(1) is Q. That is, you can actually travel continously from any point in a path-connected space to any other.
Path-connectedness is a stronger property than connectedness. A connected space is not the union of disjoint separable subsets. There exist spaces that are connected without being path-connected. An example is that formed by uniting the topologist's sine curve (q.v.) with its limit interval.