Definition 2

A topological space B is said to be unloopable if it is locally arcwise connected and if every point $b$ of B possesses a neighbourhood V such that the homomorphism of $\pi_1(V, b)$ into $\pi_1(B, b)$ deduced from the canonical injection has as image the identity element.

A locally arcwise connected topological space is unloopable if and only if each of its connected components is so.

Let B be a locally arcwise connected topological space. Suppose that every point of B possesses a neighbourhood V which is unloopable. Then B is unloopable.

Every topological space which is locally arcwise connected and arcwise simply connected is unloopable. This is in particular the case for a locally arcwise connected space homotopic to a point.