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One definition of continuity is that the pre-image of an open set is open. Can I define Continuity as that the image of an open set is open AND the image of close set is close? If this is not correct, are there any counterexamples?

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  • $\begingroup$ What you are asking is if an open/closed map is necessarily continuous. Consider the floor function $\lfloor\cdot\rfloor:\mathbb{R} \to \mathbb{Z}$ where $\mathbb{Z}$ has the discrete topology. $\endgroup$ – anakhro Aug 13 '15 at 13:16
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Hint: Consider the floor function $\lfloor\cdot\rfloor:\mathbb{R} \to \mathbb{Z}$ where $\mathbb{Z}$ is endowed with the discrete topology.

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