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[Merged by Bors] - feat: two missing lemmas about restricting continuous maps #6616

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[Merged by Bors] - feat: two missing lemmas about restricting continuous maps #6616

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feat: two missing lemmas about restricting continuous maps
ADedecker Aug 16, 2023
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ADedecker Aug 16, 2023
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ADedecker Aug 19, 2023
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15 changes: 13 additions & 2 deletions Mathlib/Topology/Constructions.lean
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Expand Up @@ -1099,9 +1099,10 @@ theorem closure_subtype {x : { a // p a }} {s : Set { a // p a }} :
closure_induced
#align closure_subtype closure_subtype

@[simp]
theorem continuousAt_codRestrict_iff {f : α → β} {t : Set β} (h1 : ∀ x, f x ∈ t) {x : α} :
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ContinuousAt (codRestrict f t h1) x ↔ ContinuousAt f x := by
simp_rw [inducing_subtype_val.continuousAt_iff, Function.comp, val_codRestrict_apply]
ContinuousAt (codRestrict f t h1) x ↔ ContinuousAt f x :=
inducing_subtype_val.continuousAt_iff
#align continuous_at_cod_restrict_iff continuousAt_codRestrict_iff

alias continuousAt_codRestrict_iff ↔ _ ContinuousAt.codRestrict
Expand All @@ -1123,6 +1124,16 @@ theorem Continuous.codRestrict {f : α → β} {s : Set β} (hf : Continuous f)
hf.subtype_mk hs
#align continuous.cod_restrict Continuous.codRestrict

@[continuity]
theorem Continuous.restrict {f : α → β} {s : Set α} {t : Set β} (h1 : MapsTo f s t)
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(h2 : Continuous f) : Continuous (h1.restrict f s t) :=
(h2.comp continuous_subtype_val).codRestrict _

@[continuity]
theorem Continuous.restrictPreimage {f : α → β} {s : Set β} (h : Continuous f) :
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Continuous (s.restrictPreimage f) :=
h.restrict _

theorem Inducing.codRestrict {e : α → β} (he : Inducing e) {s : Set β} (hs : ∀ x, e x ∈ s) :
Inducing (codRestrict e s hs) :=
inducing_of_inducing_compose (he.continuous.codRestrict hs) continuous_subtype_val he
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