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

Does there exist a **compact subspace A⊂X** that meets every equivalence class of ∼?

(This would then imply that **A /∼** is **homeomorphic to X /∼**).