Adjective
path-connected (not comparable)
(of a topological space) Such that every pair of points in the space comprises the boundary of some path mapped to the space continuously.
If X is path-connected, then this homomorphism is surjective and its kernel is the commutator subgroup of π 1 (X, x 0 ), and H 1 (X) is therefore isomorphic to the abelianization of π 1 (X, x 0 ). Source: Internet
Path connectedness This subspace of R² is path-connected, because a path can be drawn between any two points in the space. Source: Internet
A locally path-connected space is connected if and only if it is path-connected. Source: Internet
Every path-connected space is connected. Source: Internet
Every ultra-connected space is path-connected. Source: Internet
Geometric interpretation Conjugacy classes in the fundamental group of a path-connected topological space can be thought of as equivalence classes of free loops under free homotopy. Source: Internet