Past exam of the options II course of the University of Oxford / 2007 / E / E4 / a

For the surjection $p$ to the set of equivalence classes, define a subset $V$ of the quotient to be open exactly when $p^{-1}(V)$ is open in $X$. Inverse images preserve the empty and full sets, arbitrary unions and finite intersections. Applying the topology axioms in $X$ proves this is a topology on the quotient, and it makes $p$ continuous.

The map need not be open. On $X=\mathbb{R}$, identify zero with one and leave every other point in its own class. The set $U=(-1/4,1/4)$ is open, but its saturation iswhich is not open in $\mathbb{R}$. Thus $p(U)$ is not open in the quotient topology. Continuity of a quotient map does not imply openness.