Example

Here are two examples in which the hypotheses of theorem 1 are satisfied.

1) Let Y be an unlaceable topological space. Let $(A_i)_{i\in I}$ be a locally finite covering of Y by closed sets. Suppose that, for every $i\in I$, the space $A_i$ is unlaceable and that, for every pair $(i, j)\in I\times I$, the space $A_i\cap A_j$ is locally path-connected. One may take for X the sum space of the family $(A_i)_{i\in I}$ and for $f: X\rightarrow Y$ the mapping deduced from the family of canonical injections.

2) Let G be a discrete group operating properly in an unlaceable topological space X, put $Y = X/G$ and let $f: X\rightarrow Y$ be the canonical mapping. It is open (TG, III, p. 10, lemma 2) and has the path-lifting property by virtue of Theorem 4 of III, p. 287. The space Y is unlaceable by IV, p. 349, Prop. 8, b). By hypothesis, the mapping from $G\times X$ into $X\times X$ given by $(g, x)\mapsto (g\cdot x, x)$ is proper, hence strict (I, p. 18, Example 2) and its image is $X\times_YX$. It then follows from III, p. 261, Prop. 8 that $X\times_YX$ is locally path-connected.