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We know that an invertible continuous function whose inverse is also continuous is called a homeomorphism. But is there a name for a not-necessarily-bijective function that is "bicontinuous" in the sense that it sends open sets to open sets and the preimage of open sets are open sets?

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The term you're looking for is an open (continuous) map. Usually continuity is assumed, and the word is suppressed from notation.

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A function with the first property is called open (or an open mapping) and the second is just continuity, so I would call this an open continuous function.

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