= Collapsing a subset to a point
{title2=$q_A:X\to X/A$}
For a nonempty <subset> $A$ of a <topological space> $X$, identify all points of $A$ and leave each point of $X\setminus A$ in its own <equivalence class>. The resulting <quotient space> is denoted $X/A$. For every <subset> $K\subseteq X$, its <saturation of a subset> is
$$
q_A^{-1}(q_A(K))=
\begin{cases}
K,&K\cap A=\varnothing,\\
K\cup A,&K\cap A\ne\varnothing.
\end{cases}
$$
The <quotient topology> therefore gives a useful test: $q_A(K)$ is a <closed set> if and only if the displayed <saturation of a subset> is a <closed set> of $X$. In particular, every <closed set> disjoint from $A$ has <closed> image, even if $A$ is not <closed>. When $A$ is <closed>, $q_A$ is a <closed map>, because both possibilities in the formula are then <closed sets> for <closed> $K$.
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