Let X and Y be finite topological spaces and let f:X \to Y be continuous. Show that if the inverse image of the closure of each point in Y is contractible (or has the weak homotopy type of a point), then f is a weak homotopy equivalence.
(Question: for finite spaces, does weak homotopy equivalent to a point imply contractible? This is not true for arbitrary "locally finite" spaces (eg the universal covering space of the non-simply connected 4-point space), but it feels more likely to be true for finite spaces. If I ever figure out how to prove or disprove it (or if one of you, dear readers, does so and tells me about it!), it'll be a FLMPotD.)
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FLMPotD (Quillen Fiber Lemma)
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