Proposition 8

Every quotient space of a locally path-connected space is locally path-connected.

Let X be a locally path-connected space, let R be an equivalence relation in X and let $\varphi : X\rightarrow X/R$ be the canonical mapping. It suffices to prove that the path-connected components of an open subset of $X/R$ are open. Let thus A be an open subset of $X/R$ and let C be a path-connected component of A. Let $x\in \overset{-1}{\varphi}(C)$, and let K be the path-connected component of $x$ in $\overset{-1}{\varphi}(A)$. The set $\varphi (K)$ is path-connected (III, p. 258, prop. 3), contained in A and contains $\varphi (x)$; hence $\varphi (K)\subset C$, whence $K\subset \overset{-1}{\varphi}(C)$. This proves that $\overset{-1}{\varphi}(C)$ is the union of path-connected components of $\overset{-1}{\varphi}(A)$. Since X is locally path-connected and $\overset{-1}{\varphi}(A)$ is open in X$,\overset{-1}{\varphi}(C)$ is open in X. Consequently, C is open in $X/R$. This proves the proposition.