= 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:
$$
p^{-1}(p(U))=U.
$$
Its preimage is open, so the <quotient topology> gives
$$
\boxed{p(X\setminus K)\text{ is open in }X_A.}
$$
The hypothesis $K\cap A=\varnothing$ is exactly what prevents the <saturation of a subset> from adding missing points of the collapsed class.
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