Proposition 4

Let B be an unwindable topological space. The relation of strict homotopy is an open equivalence relation in the topological space $\mathscr{C}_c(\mathbf{I}; B)$.

Suppose first that the space B is simply arcwise connected. Let $\varphi : \Lambda (B)\rightarrow B\times B$ denote the mapping which associates with a path $c$ in B the pair $(c(0), c(1))$ formed by its origin and its end. In order that two paths in B be strictly homotopic, it is necessary and sufficient that they have the same origin and the same end (III, p. 292, prop. 2). Since the space B is connected and locally arcwise connected, the mapping $\varphi$ is surjective, continuous and open (III, p. 262, prop. 10). The relation of strict homotopy is therefore open (TG, I, p. 32, prop. 3).

In the general case, let E be a simply connected by paths covering of B, nonempty if B $=\not\emptyset$. This covering is surjective because B is connected (I, p. 74, prop. 4). Let us denote by $p$ the projection of the B-space E and let $\widetilde{p}: \Lambda (E)\rightarrow \Lambda (B)$ be the mapping which associates to a path $c$ in E the path $p\circ c$. Endowed with the mapping $\widetilde{p}$, Λ(E) is a covering of Λ(B) (III, p. 302, cor. 1 of prop. 3). In order that two paths in B should be strictly homotopic, it is necessary and sufficient that they should be the images under the mapping $\widetilde{p}$ of two strictly homotopic paths in E (III, p. 302, prop. 4). Let U be an open subset of the space Λ(B). The set $(\overset{-1}{\widetilde{p}})(U)$ is open in Λ(E) ; by the first part of the

proof, the set $U'$ saturated of $(\overset{-1}{\widetilde{p}})(U)$ for the relation of strict homotopy is open in Λ(E). Since Λ(E) is a covering of Λ(B), the set $\widetilde{p}(U')$ is open in Λ(B). This proves the proposition, for $\widetilde{p}(U')$ is the saturated set of U for the relation of strict homotopy.