Proposition 12

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

Let $X$ be a locally connected space, $R$ an equivalence relation on $X$, $\varphi : X \to X/R$ the canonical mapping. Let $A$ be an open subset of

X/R and C a component of A. Then $\overline{\varphi}^1(C)$ is a union of components of $\overline{\varphi}^1(A)$; for if $x \in \overline{\varphi}^1(C)$ and if K is the component of x in $\overline{\varphi}^1(A)$ then $\varphi(K)$ is connected (no. 2, Proposition 4), is contained in A, and contains $\varphi(x)$; hence $\varphi(K) \subset C$ by the definition of C, and therefore $K \subset \overline{\varphi}^1(C)$. Since X is locally connected and $\overline{\varphi}^1(A)$ is open in X, it follows from Proposition 11 that $\overline{\varphi}^1(C)$ is open in X; consequently C is open in X/R and hence, by Proposition 11 again, X/R is locally connected.