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refactor(topology/algebra/ordered): reduce imports (#7601)
Renames `topology.algebra.ordered` to `topology.algebra.order`, and moves the material about `liminf/limsup` and about `extend_from` to separate files, in order to delay imports. Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
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/- | ||
Copyright (c) 2017 Johannes Hölzl. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov | ||
-/ | ||
import topology.algebra.ordered.basic | ||
import topology.extend_from | ||
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/-! | ||
# Lemmas about `extend_from` in an order topology. | ||
-/ | ||
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open filter set topological_space | ||
open_locale topological_space classical | ||
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universes u v | ||
variables {α : Type u} {β : Type v} | ||
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lemma continuous_on_Icc_extend_from_Ioo [topological_space α] [linear_order α] [densely_ordered α] | ||
[order_topology α] [topological_space β] [regular_space β] {f : α → β} {a b : α} | ||
{la lb : β} (hab : a < b) (hf : continuous_on f (Ioo a b)) | ||
(ha : tendsto f (𝓝[Ioi a] a) (𝓝 la)) (hb : tendsto f (𝓝[Iio b] b) (𝓝 lb)) : | ||
continuous_on (extend_from (Ioo a b) f) (Icc a b) := | ||
begin | ||
apply continuous_on_extend_from, | ||
{ rw closure_Ioo hab, }, | ||
{ intros x x_in, | ||
rcases mem_Ioo_or_eq_endpoints_of_mem_Icc x_in with rfl | rfl | h, | ||
{ use la, | ||
simpa [hab] }, | ||
{ use lb, | ||
simpa [hab] }, | ||
{ use [f x, hf x h] } } | ||
end | ||
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lemma eq_lim_at_left_extend_from_Ioo [topological_space α] [linear_order α] [densely_ordered α] | ||
[order_topology α] [topological_space β] [t2_space β] {f : α → β} {a b : α} | ||
{la : β} (hab : a < b) (ha : tendsto f (𝓝[Ioi a] a) (𝓝 la)) : | ||
extend_from (Ioo a b) f a = la := | ||
begin | ||
apply extend_from_eq, | ||
{ rw closure_Ioo hab, | ||
simp only [le_of_lt hab, left_mem_Icc, right_mem_Icc] }, | ||
{ simpa [hab] } | ||
end | ||
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lemma eq_lim_at_right_extend_from_Ioo [topological_space α] [linear_order α] [densely_ordered α] | ||
[order_topology α] [topological_space β] [t2_space β] {f : α → β} {a b : α} | ||
{lb : β} (hab : a < b) (hb : tendsto f (𝓝[Iio b] b) (𝓝 lb)) : | ||
extend_from (Ioo a b) f b = lb := | ||
begin | ||
apply extend_from_eq, | ||
{ rw closure_Ioo hab, | ||
simp only [le_of_lt hab, left_mem_Icc, right_mem_Icc] }, | ||
{ simpa [hab] } | ||
end | ||
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lemma continuous_on_Ico_extend_from_Ioo [topological_space α] | ||
[linear_order α] [densely_ordered α] [order_topology α] [topological_space β] | ||
[regular_space β] {f : α → β} {a b : α} {la : β} (hab : a < b) (hf : continuous_on f (Ioo a b)) | ||
(ha : tendsto f (𝓝[Ioi a] a) (𝓝 la)) : | ||
continuous_on (extend_from (Ioo a b) f) (Ico a b) := | ||
begin | ||
apply continuous_on_extend_from, | ||
{ rw [closure_Ioo hab], exact Ico_subset_Icc_self, }, | ||
{ intros x x_in, | ||
rcases mem_Ioo_or_eq_left_of_mem_Ico x_in with rfl | h, | ||
{ use la, | ||
simpa [hab] }, | ||
{ use [f x, hf x h] } } | ||
end | ||
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lemma continuous_on_Ioc_extend_from_Ioo [topological_space α] | ||
[linear_order α] [densely_ordered α] [order_topology α] [topological_space β] | ||
[regular_space β] {f : α → β} {a b : α} {lb : β} (hab : a < b) (hf : continuous_on f (Ioo a b)) | ||
(hb : tendsto f (𝓝[Iio b] b) (𝓝 lb)) : | ||
continuous_on (extend_from (Ioo a b) f) (Ioc a b) := | ||
begin | ||
have := @continuous_on_Ico_extend_from_Ioo (order_dual α) _ _ _ _ _ _ _ f _ _ _ hab, | ||
erw [dual_Ico, dual_Ioi, dual_Ioo] at this, | ||
exact this hf hb | ||
end |
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