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 /∼).