[Merged by Bors] - feat(topology/algebra/ordered): monotone continuous function is homeomorphism, relative version #4043
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Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
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…morphism, relative version (#4043) A function `f : α → β` restricts to a homeomorphism `(Ioo a b) → β`, if it (1) is order-preserving within the interval; (2) is `continuous_on` the interval; (3) tends to positive infinity at the right endpoint; and (4) tends to negative infinity at the left endpoint. The orders `α`, `β` are required to be conditionally complete, and the order on `α` must also be dense.
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A function
f : α → βrestricts to a homeomorphism(Ioo a b) → β, if it (1) is order-preserving within the interval; (2) iscontinuous_onthe interval; (3) tends to positive infinity at the right endpoint; and (4) tends to negative infinity at the left endpoint. The ordersα,βare required to be conditionally complete, and the order onαmust also be dense.This is a relative version of #3843. As explained in a docstring, it is only one of dozens of possible relative versions -- one can vary the type of interval in both the source and target. I provide this one as a proof of concept (building up the infrastructure for any of the others should be possible with this as a model), and because it is needed for a future PR of @benjamindavidson.
Zulip discussion of one of the lemmas
Blocked by #3991.