 Collapsing a subset to a point

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 isThe 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 $K$.