Proposition 7

Let $X$ be a topological space and $R$ an equivalence relation on $X$. If the quotient space $X/R$ is connected, and if each equivalence class mod $R$ is connected, then $X$ is connected.

Suppose $X$ is not connected. Then there is a partition of $X$ into two open sets $A, B$. The sets $A, B$ are saturated with respect to $R$; for if $x \in A$ then the equivalence class $M$ of $x$ cannot meet $B$, otherwise the sets $A \cap M, B \cap M$ would form a partition of $M$ into two sets open in $M$, which is impossible since $M$ is connected. The canonical images of $A$ and $B$ are therefore open sets in $X/R$ and form a partition of $X/R$; this contradicts the assumption that $X/R$ is connected.