Example 1

Let X be a topological space and let R be an equivalence relation in X. Let Y denote the quotient topological space $X/R$ and $f: X\rightarrow Y$ the canonical mapping. Suppose that the equivalence classes of R are connected. Then, for every open and

connected subset V of Y, the set $\overset{-1}{f}(V)$ is connected (TG, I, p. 23, corollary 1 and p. 82, proposition 7). If the space Y is locally connected, the mapping $f$ is thus relatively connected.