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@boarbarktree @ClaudioJacobo @skittishLilly And given any example f:X→X, here's a new compact non-Hausdorff example: assume z in XU{z} only has the whole space as a neighborhood and let f(z)=z.
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@boarbarktree @ClaudioJacobo @skittishLilly Hmm, well just "maybe new". You could do this same gimmick with an infinite set Z, but then the above procedure produces a homeomorphic copy.