[Merged by Bors] - refactor(order/filter,topology): review API #6347
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### Filters * move old `filter.map_comap` to `filter.map_comap_of_mem`; * add a new `filter.map_comap` that doesn't assume `range m ∈ f` but gives `map m (comap m f) = f ⊓ 𝓟 (range m)` instead of `map m (comap m f) = f`; * add `filter.comap_le_comap_iff`, use it to golf a couple of proofs; * move some lemmas using `filter.map_comap`/`filter.map_comap_of_mem` closure to these lemmas; * use `function.injective m` instead of `∀ x y, m x = m y → x = y` in some lemmas; * drop `filter.le_map_comap_of_surjective'`, `filter.map_comap_of_surjective'`, and `filter.le_map_comap_of_surjective`: the inequalities easily follow from equalities, and `filter.map_comap_of_surjective'` is just `filter.map_comap_of_mem`+`filter.mem_sets_of_superset`; ### Topology * add `closed_embedding_subtype_coe`, `ennreal.open_embedding_coe`; * drop `inducing_open`, `inducing_is_closed`, `embedding_open`, and `embedding_is_closed` replace them with `inducing.is_open_map` and `inducing.is_closed_map`; * move old `inducing.map_nhds_eq` to `inducing.map_nhds_of_mem`, the new `inducing.map_nhds_eq` says `map f (𝓝 a) = 𝓝[range f] (f a)`; * add `inducing.is_closed_iff`; * move old `embedding.map_nhds_eq` to `embedding.map_nhds_of_mem`, the new `embedding.map_nhds_eq` says `map f (𝓝 a) = 𝓝[range f] (f a)`; * add `open_embedding.map_nhds_eq`; * change signature of `is_closed_induced_iff` to match other similar lemmas; * move old `map_nhds_induced_eq` to `map_nhds_induced_of_mem`, the new `map_nhds_induced_eq` give `𝓝[range f] (f a)` instead of `𝓝 (f a)`.
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### Filters * move old `filter.map_comap` to `filter.map_comap_of_mem`; * add a new `filter.map_comap` that doesn't assume `range m ∈ f` but gives `map m (comap m f) = f ⊓ 𝓟 (range m)` instead of `map m (comap m f) = f`; * add `filter.comap_le_comap_iff`, use it to golf a couple of proofs; * move some lemmas using `filter.map_comap`/`filter.map_comap_of_mem` closure to these lemmas; * use `function.injective m` instead of `∀ x y, m x = m y → x = y` in some lemmas; * drop `filter.le_map_comap_of_surjective'`, `filter.map_comap_of_surjective'`, and `filter.le_map_comap_of_surjective`: the inequalities easily follow from equalities, and `filter.map_comap_of_surjective'` is just `filter.map_comap_of_mem`+`filter.mem_sets_of_superset`; ### Topology * add `closed_embedding_subtype_coe`, `ennreal.open_embedding_coe`; * drop `inducing_open`, `inducing_is_closed`, `embedding_open`, and `embedding_is_closed` replace them with `inducing.is_open_map` and `inducing.is_closed_map`; * move old `inducing.map_nhds_eq` to `inducing.map_nhds_of_mem`, the new `inducing.map_nhds_eq` says `map f (𝓝 a) = 𝓝[range f] (f a)`; * add `inducing.is_closed_iff`; * move old `embedding.map_nhds_eq` to `embedding.map_nhds_of_mem`, the new `embedding.map_nhds_eq` says `map f (𝓝 a) = 𝓝[range f] (f a)`; * add `open_embedding.map_nhds_eq`; * change signature of `is_closed_induced_iff` to match other similar lemmas; * move old `map_nhds_induced_eq` to `map_nhds_induced_of_mem`, the new `map_nhds_induced_eq` give `𝓝[range f] (f a)` instead of `𝓝 (f a)`.
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Filters
filter.map_comaptofilter.map_comap_of_mem;filter.map_comapthat doesn't assumerange m ∈ fbutgives
map m (comap m f) = f ⊓ 𝓟 (range m)instead ofmap m (comap m f) = f;filter.comap_le_comap_iff, use it to golf a couple of proofs;filter.map_comap/filter.map_comap_of_memclosure to these lemmas;
function.injective minstead of∀ x y, m x = m y → x = yinsome lemmas;
filter.le_map_comap_of_surjective',filter.map_comap_of_surjective', andfilter.le_map_comap_of_surjective: the inequalities easily followfrom equalities, and
filter.map_comap_of_surjective'is justfilter.map_comap_of_mem+filter.mem_sets_of_superset;Topology
closed_embedding_subtype_coe,ennreal.open_embedding_coe;inducing_open,inducing_is_closed,embedding_open, andembedding_is_closedreplace them withinducing.is_open_mapandinducing.is_closed_map;inducing.map_nhds_eqtoinducing.map_nhds_of_mem, thenew
inducing.map_nhds_eqsaysmap f (𝓝 a) = 𝓝[range f] (f a);inducing.is_closed_iff;embedding.map_nhds_eqtoembedding.map_nhds_of_mem, thenew
embedding.map_nhds_eqsaysmap f (𝓝 a) = 𝓝[range f] (f a);open_embedding.map_nhds_eq;is_closed_induced_iffto match other similarlemmas;
map_nhds_induced_eqtomap_nhds_induced_of_mem, the newmap_nhds_induced_eqgive𝓝[range f] (f a)instead of𝓝 (f a).