Let <b>X</b> be <b>a topological space</b> that is not too bad (let's say "not too bad" = "compactly generated Hausdorff"), and let ∼ be an equivalence relation such that <b>X /∼ is compact Hausdorff</b>.

Does there exist a <b>compact subspace A⊂X</b> that meets every equivalence class of ∼?<br> (This would then imply that <b>A /∼</b> is <b>homeomorphic to X /∼</b>).