[Merged by Bors] - chore(topology/order): relate Sup and Inf of topologies to generate_from
#9045
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hrmacbeth
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semorrison
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semorrison
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src/topology/order.lean
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| @galois_insertion.u_le_u_iff _ _ _ _ _ | ||
| (@partial_order.to_preorder _ | ||
| (@semilattice_sup.to_partial_order _ | ||
| (@lattice.to_semilattice_sup _ | ||
| (@bounded_lattice.to_lattice _ | ||
| (@complete_lattice.to_bounded_lattice _ tmp_complete_lattice))))) | ||
| (gi_generate_from α) a b |
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| @galois_insertion.u_le_u_iff _ _ _ _ _ | |
| (@partial_order.to_preorder _ | |
| (@semilattice_sup.to_partial_order _ | |
| (@lattice.to_semilattice_sup _ | |
| (@bounded_lattice.to_lattice _ | |
| (@complete_lattice.to_bounded_lattice _ tmp_complete_lattice))))) | |
| (gi_generate_from α) a b | |
| @galois_insertion.u_le_u_iff _ (order_dual (topological_space α)) _ _ _ _ (gi_generate_from α) a b |
Hopefully the same works below?
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Thanks, yes! I've changed throughout. I'm suprised because order_dual (topological_space α) actually carries the dual of the dual of tmp_order, whereas the Galois insertion is stated for tmp_order. I had assumed that an order was not defeq to its double dual.
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…from` (#9045) Since there is a Galois insertion between `generate_from : set (set α) → topological_space α` and the "forgetful functor" `topological_space α → set (set α)`, all kinds of lemmas about the interaction of `generate_from` and the ordering on topologies automatically follow. But it is hard to use the Galois insertion lemmas directly, because the Galois insertion is actually provided for the dual order on topologies, which confuses Lean. Here we re-state most of the Galois insertion API in this special case. Co-authored-by: Johan Commelin <johan@commelin.net>
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generate_fromgenerate_from
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Since there is a Galois insertion between
generate_from : set (set α) → topological_space αand the "forgetful functor"topological_space α → set (set α), all kinds of lemmas about the interaction ofgenerate_fromand the ordering on topologies automatically follow. But it is hard to use the Galois insertion lemmas directly, because the Galois insertion is actually provided for the dual order on topologies, which confuses Lean. Here we re-state most of the Galois insertion API in this special case.