feat(topology/order/scott): Introduce the Scott topology on a preorder #18448
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Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
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src/topology/order/scott.lean
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| end, | ||
| rw with_scott_topology.is_open_eq_upper_and_lub_mem_implies_inter_nonempty at hu, | ||
| have e2: ((f '' d) ∩ u).nonempty := | ||
| begin |
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probably you can use show_term {} to turn this whole block into a 1-liner
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@alexjbest Do you mean:
have e2: ((f '' d) ∩ u).nonempty,
show_term { sorry, },
? How do I know what tatic to replace sorry with? There don't seem to be any examples of use of show_term in mathlib outside of tatics.
Co-authored-by: Alex J Best <alex.j.best@gmail.com>
Co-authored-by: Alex J Best <alex.j.best@gmail.com>
Co-authored-by: Alex J Best <alex.j.best@gmail.com>
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All prerequisites are ported. Would you mind closing this PR and reopening it on the mathlib4 side, so that we have one less file to port? |
Sure! |
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PR opened (needs more work) leanprover-community/mathlib4#2508 |
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We prove an insert result for directed sets when the relation is reflexive. This is then used to show that a Scott continuous function is monotone. This result is required in the [construction of the Scott topology on a preorder](leanprover-community/mathlib4#2508) (see also #18448). Holding PR for mathlib4: leanprover-community/mathlib4#2543 Co-authored-by: Christopher Hoskin <mans0954@users.noreply.github.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
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Introduces the Scott topology on a preorder, defined in terms of directed sets.
There is already a related notion of Scott topology defined in
topology.omega_complete_partial_order, where it is defined on ω-complete partial orders in terms of ω-chains. In some circumstances the definition given here coincides with that given intopology.omega_complete_partial_orderbut in general they are different. Abramsky and Jung ([Domain Theory, 2.2.4][abramsky_gabbay_maibaum_1994]) argue that the ω-chain approach has pedagogical advantages, but the directed sets approach is more appropriate as a theoretical foundation.