Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / E / E4 / a

A subset is connected if it has no decomposition into two disjoint nonempty sets both relatively open in it. Equivalently its only relatively clopen subsets are itself and the empty set. Connectedness here refers to the subset's induced topology.

Suppose $f(X)=U\cup V$ were a separation into two disjoint nonempty relatively open sets. Continuity makes $f^{-1}(U)$ and $f^{-1}(V)$ open in $X$, and they are disjoint, nonempty and cover $X$. This contradicts connectedness of $X$. Hence $\boxed{f(X)\text{ is connected}}$.

The set $X/R$ consists of the equivalence classes, with canonical surjection $q:x\mapsto [x]$. The quotient topology declares $V\subseteq X/R$ open exactly when $q^{-1}(V)$ is open in $X$. This makes $q$ continuous and is the finest topology on the quotient set doing so.