= Solution
For the canonical <surjection> $p:X\to X/{\sim}$, define a subset $W$ of $X/{\sim}$ to be open exactly when $p^{-1}(W)$ is open in $X$. This is the <quotient topology>. The empty set and the whole quotient have open <inverse images>. Moreover <inverse images> commute with arbitrary unions and finite intersections, so the corresponding axioms of a <topology> follow directly from those on $X$.
The map $p$ need not be an <open map>. For example, on $X=\mathbb R$ identify every point of $[0,1]$ to one point and leave all other classes singletons. The set $U=(-1/2,1/2)$ is open, but
$$
p^{-1}(p(U))=(-1/2,1]
$$
is not open. The definition of the <quotient topology> therefore shows that $p(U)$ is not open. The obstruction is that saturating an <open set> by <equivalence classes> need not preserve openness.
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