= Noncompactness after collapsing a countable real subset
{title2=$A\subseteq\mathbb R\text{ countable}\implies\mathbb R/A\text{ not compact}$}
Choose $b_n\in(n,n+1/2)\setminus A$, possible because each interval is uncountable. Every tail $K_n=\{b_k:k\geq n\}$ 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.
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