 Using this set, a space is 0-connected if and only if the 0th homotopy set is the one-point set.The definition of homotopy groups and this homotopy set require that "X" be pointed (have a chosen base point), which cannot be done if "X" is empty.A topological space "X" is path-connected if and only if its 0-th homotopy group vanishes identically, as path-connectedness implies that any two points "x1" and "x2" in "X" can be connected with a continuous path which starts in "x1" and ends in "x2", which is equivalent to the assertion that every mapping from "S0" (a discrete set of two points) to "X" can be deformed continuously to a constant map.