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refactor(topology/{order,*}): review API (#18312)
## Main changes * Switch from `@[class] structure topological_space` to `class`. * Introduce notation `is_open[t]`/`is_closed[t]`/`𝓤[u]` and use it instead of `t.is_open`/`@is_closed _ t`/`u.uniformity` * Don't introduce a temporary order on `topological_space`, use `galois_coinsertion` to the order-dual of `set (set α)` instead. * Drop `discrete_topology_bot`. It seems that this instance doesn't work well in Lean 4 (in fact, I failed to define it without using `@DiscreteTopology.mk`). ## Other changes ### Topological spaces * Add `is_open_mk`. * Rename `generate_from_mono` to `generate_from_anti`, move it to the `topological_space` namespace. * Add `embedding_inclusion`, `embedding_ulift_down`, and `ulift.discrete_topology`. * Move `discrete_topology.of_subset` from `topology.separation` to `topology.constructions`. * Move `embedding.discrete_topology` from `topology.separation` to `topology.maps`. * Use explicit arguments in an argument of `nhds_mk_of_nhds`. * Move some definitions and lemmas like `mk_of_closure`, `gi_generate_from` (renamed to `gci_generate_from`), `left_inverse_generate_from` to the `topological_space` namespace. These lemmas are very specific and use generic names. * Add `topological_space` and `discrete_topology` instances for `fin n`. * Drop `is_open_induced_iff'`, use non-primed version instead. * Prove `set_of_is_open_sup` by `rfl`. * Drop `nhds_bot`, use `nhds_discrete` instead. * Drop `induced_bot` and `discrete_topology_induced` ### Uniform spaces * Drop `infi_uniformity'` and `inf_uniformity'`. * Use `@uniformity α (uniform_space.comap _ _)` instead of `(h : ‹uniform_space α› = uniform_space.comap f ‹uniform_space β›)` in `uniformity_comap`. ### Locally constant functions and discrete quotients * Use quotient space topology for the coercion of a `discrete_quotient` to `Type*`, then prove `[discrete_topology]`. This way we avoid surprising diamonds in the future (especially if Lean 4 will unfold the coercion). * Merge `locally_constant.lift` and `locally_constant.locally_constant_lift` into 1 def, rename `factors` to `lift_comp_proj`. * Protect `locally_constant.is_locally_constant`. * Drop `locally_constant.iff_continuous_bot` ### Categories * Add an instance for `discrete_topology (discrete.obj X)`. * Rename `Fintype.discrete_topology` to `Fintype.bot_topology `, add lemma `Fintype.discrete_topology` sating that this is a `[discrete_topology]`. ### Topological rings * Fix&golf a proof that failed because of API changes.
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