Corollary

Soit X a topological space locally connected by arcs and let $x$ be a point of X. The mapping $c\mapsto c(1)$ of $\Lambda_x(X)$ into X is open.

According to the proposition, the mapping $\varphi : \Lambda (X)\rightarrow X\times X$ defined by $\varphi (c) = (c(0), c(1))$ is open and one has $\Lambda_x(X) =\overset{-1}{\varphi}(\{x\} \times X)$. Consequently, the mapping $\Lambda_x(X)\rightarrow \{x\} \times X$ deduced from $\varphi$ is open (TG, I, p. 30, prop. 2), as is its composite with the second projection pr$_2$.