[Merged by Bors] - refactor(topology/bases): use structure for `is_topological_basis
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* add `set.range_prod_map` and `set.countable.image2`; * move `@[simp]` from `set.mem_image2` to `set.mem_image2_eq` so that `dsimp` can use it; * rename `set.countable_prod` to `set.countable.prod`.
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| ⟨λ ha, let ⟨b, hb, ab, bu⟩ := mem_basis_subset_of_mem_open hB ha ou in | ||
| ⟨b, ⟨hb, bu⟩, ab⟩, | ||
| λ ⟨b, ⟨hb, bu⟩, ab⟩, bu ab⟩⟩ | ||
| ⟨{s ∈ B | s ⊆ u}, λ s h, h.1, hB.open_eq_sUnion' ou⟩ | ||
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| lemma Union_basis_of_is_open {B : set (set α)} |
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…6947) * turn `topological_space.is_topological_basis` into a `structure`; * rename `mem_nhds_of_is_topological_basis` to `is_topological_basis.mem_nhds_iff`; * rename `is_open_of_is_topological_basis` to `is_topological_basis.is_open`; * rename `mem_basis_subset_of_mem_open` to `is_topological_basis.exists_subset_of_mem_open`; * rename `sUnion_basis_of_is_open` to `is_topological_basis.open_eq_sUnion`, add prime version; * add `is_topological_basis.prod`; * add convenience lemma `is_topological_basis.second_countable_topology`; * rename `is_open_generated_countable_inter` to `exists_countable_basis`; * add `topological_space.countable_basis` and API around it; * add `@[simp]` to `opens.mem_supr`, add `opens.mem_Sup`; * golf
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structure for `is_topological_basisstructure for `is_topological_basis
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…6947) * turn `topological_space.is_topological_basis` into a `structure`; * rename `mem_nhds_of_is_topological_basis` to `is_topological_basis.mem_nhds_iff`; * rename `is_open_of_is_topological_basis` to `is_topological_basis.is_open`; * rename `mem_basis_subset_of_mem_open` to `is_topological_basis.exists_subset_of_mem_open`; * rename `sUnion_basis_of_is_open` to `is_topological_basis.open_eq_sUnion`, add prime version; * add `is_topological_basis.prod`; * add convenience lemma `is_topological_basis.second_countable_topology`; * rename `is_open_generated_countable_inter` to `exists_countable_basis`; * add `topological_space.countable_basis` and API around it; * add `@[simp]` to `opens.mem_supr`, add `opens.mem_Sup`; * golf
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topological_space.is_topological_basisinto astructure;mem_nhds_of_is_topological_basistois_topological_basis.mem_nhds_iff;is_open_of_is_topological_basistois_topological_basis.is_open;mem_basis_subset_of_mem_opentois_topological_basis.exists_subset_of_mem_open;sUnion_basis_of_is_opentois_topological_basis.open_eq_sUnion, add prime version;is_topological_basis.prod;is_topological_basis.second_countable_topology;is_open_generated_countable_intertoexists_countable_basis;topological_space.countable_basisand API around it;@[simp]toopens.mem_supr, addopens.mem_Sup;set.countable.prod