[Merged by Bors] - refactor(topology/bases): removing/adding the empty set from/to a basis #17133
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Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>
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…is (#17133) + Add lemmas `is_topological_basis.insert/diff_empty` that allows adding the empty set to a basis and removing the empty set from a basis. + Remove `(⋂₀ f).nonempty` from `is_topological_basis_of_subbasis` and golf the proof. This condition is unnatural and was only used to exclude the empty set in [topological_space.second_countable_topology.to_separable_space](https://leanprover-community.github.io/mathlib_docs/topology/bases.html#topological_space.second_countable_topology.to_separable_space), because we want to choose a point from each set in the basis. As a result, the proof `exists_countable_basis` needs a slight modification.
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Add lemmas
is_topological_basis.insert/diff_emptythat allows adding the empty set to a basis and removing the empty set from a basis.Remove
(⋂₀ f).nonemptyfromis_topological_basis_of_subbasisand golf the proof. This condition is unnatural and was only used to exclude the empty set in topological_space.second_countable_topology.to_separable_space, because we want to choose a point from each set in the basis. As a result, the proofexists_countable_basisneeds a slight modification.