[Merged by Bors] - feat(Topology/Order): Sierpiński space classifies open sets of Scott topology#16826
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[Merged by Bors] - feat(Topology/Order): Sierpiński space classifies open sets of Scott topology#16826
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PR summary e9366d1ba0Import changes for modified filesNo significant changes to the import graph Import changes for all files
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Co-authored-by: Yaël Dillies <[email protected]>
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I pushed a golf. The rest looks good
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…topology (#16826) This PR shows that the Sierpiński topology coincides with the upper topology on `Prop`, and hence with the Scott topology (since `Prop` is a complete linear order). This observation reveals that a subset of a preorder is open in the Scott topology if and only if its characteristic function is Scott continuous. Co-authored-by: Christopher Hoskin <[email protected]> Co-authored-by: Yaël Dillies <[email protected]>
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…ogy (#16523) Unify Scott Topologies, following on from #13201 and #16826. Originally Mathlib had a notion of ω-Scott Continuity and ω-Scott Topology based on Chains for ω-complete partial orders. Previously we introduced a notion of Scott Continuity and Scott Topology based on directed sets for arbitrary pre-orders (#2508). In #13201, we generalised our definition of Scott Continuity to include ω-Scott Continuity as a special case. This PR continues that work by generalising our notion of Scott Topology to include ω-Scott Topology as a special case. `Topology.IsScott.scottContinuous_iff_continuous` and `Topology.IsScott.ωscottContinuous_iff_continuous` have almost identical proofs and are presumably special cases of a more general result that I have not yet formulated. However I think this PR covers enough ground to be worth considering at this stage. Co-authored-by: Christopher Hoskin <[email protected]> Co-authored-by: Christopher Hoskin <[email protected]>
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This PR shows that the Sierpiński topology coincides with the upper topology on
Prop, and hence with the Scott topology (sincePropis a complete linear order).This observation reveals that a subset of a preorder is open in the Scott topology if and only if its characteristic function is Scott continuous.