Definition 1

A topological space X is said to be arcwise simply connected if it is arcwise connected and if every loop in X is strictly homotopic to a constant loop.

The empty space is arcwise simply connected. In order that an arcwise connected space X be arcwise simply connected, it is necessary and sufficient that the group $\pi_1(X, x)$ be reduced to the identity element for every point $x$ of X (III, p. 292, prop. 2). It is sufficient that this be so for one point $x$ of X (loc. cit.).

Every topological space homotopic to an arcwise simply connected space is arcwise simply connected. In fact, it is arcwise connected (III, p. 260) and, at every point, its Poincaré group is reduced to the identity element (III, p. 296, cor. 2 of prop. 6). In particular, a topological space homotopic to a point is arcwise simply connected.

A topological space which is arcwise simply connected and locally arcwise connected is simply connected (III, p. 309, cor. 2 of prop. 1).