= Solution
A <compact space> is a <topological space> in which every <open cover> has a finite subcover. A <Hausdorff space> is one in which every two distinct points have disjoint open neighbourhoods. A <homeomorphism> is a bijection that is continuous and whose inverse is continuous.
For an <equivalence relation> $R$ on $X$, let $X/R$ be the set of equivalence classes and let
$$
q:X\longrightarrow X/R,
\qquad q(x)=[x].
$$
The <quotient topology> declares $U\subseteq X/R$ open exactly when $q^{-1}(U)$ is open in $X$. It follows directly from the definition that $q$ is continuous.
Suppose that the continuous map $f:X\to Y$ is constant on equivalence classes. The only possible factorisation is
$$
F:X/R\longrightarrow Y,
\qquad F([x])=f(x),
$$
which is well-defined by the hypothesis and satisfies $F\circ q=f$. For every open $V\subseteq Y$,
$$
q^{-1}(F^{-1}(V))=f^{-1}(V)
$$
is open in $X$. The definition of the quotient topology therefore makes $F^{-1}(V)$ open, so $F$ is continuous. This proves the <universal property of the quotient topology>.
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