Collapsing a subset to a point ($q_A:X\to X/A$)

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 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 $K$.

Choose $b_n\in (n,n+1/2)\setminus A$, possible because each interval is uncountable. Every tail $K_n=\{b_k:k\ge n\}$ is closed and disjoint from $A$. Therefore $U_n=\mathbb{R}\setminus K_n$ is an open saturated set for collapsing a subset to a point, so its image in the quotient topology is open. These increasing images cover the quotient, but no finite selection covers it, because the largest selected image still misses an infinite tail of distinct singleton classes. This is an open cover with no finite subcover, proving noncompactness. The argument does not require the countable subset to be closed or nondense. For the empty subset, read the quotient as the original line.