Corollary 2

Let B be a locally path-connected topological space and let E be a B-space étalé. The space E is locally path-connected. If it is connected, it is path-connected.

The first assertion follows immediately from def. 4 and the definition of an étale mapping (I, p. 28, def. 2). The second follows from it by prop. 7.

In order that a topological space X be locally path-connected, it is necessary and sufficient that every path-connected component of an open set of X be an open set of X (cf. I, p. 85, proof of prop. 11). If the space X is locally path-connected, every point of X therefore possesses a fundamental system of open path-connected neighbourhoods.