 Solution

Since $K$ misses $A$, the open set $U=X\setminus K$ contains the whole collapsed equivalence class $A$. Every other equivalence class is a singleton, so $U$ is saturated under the saturation of a subset operation:Its preimage is open, so the quotient topology givesThe hypothesis $K\cap A=\varnothing$ is exactly what prevents the saturation of a subset from adding missing points of the collapsed class.