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The question is:

$f: X \to Y$ is continuous bijection, which of the following is correct:

I. if $X$ is Hausdorff space then $Y$ is Hausdorff space.

II. if $X$ is compact and $Y$ is Hausdorff space, then $f^{-1}$ exist.

I think I is correct since $X$, the Hausdorff is separable, then image should be separable. I'm not sure about II.

Besides, can you show me some example and counterexample of Hausdorff space? Thank you.

  • 1
    Since $f$ is a bijection, $f^{-1}$ exists. Nothing else is needed for II.2012-11-09

1 Answers 1