Exercise 8, § 4

Let $(G_i)_{i \in I}$ be a family of graphs of equivalence relations (Set Theory, Chapter II, § 6, no. 1) on a topological space $X$, and let $R_i$ be the relation $(x, y) \in G_i$; then

is the graph of an equivalence relation $R$ on $X$. Show that there is a canonical continuous injection of the quotient space $X/R$ into the product space $\prod_{i \in I} (X/R_i)$.