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feat: define
RestrictGenTopology
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/- | ||
Copyright (c) 2024 Yury Kudryashov. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Yury Kudryashov | ||
-/ | ||
import Mathlib.Topology.Defs.Sequences | ||
import Mathlib.Topology.Compactness.Compact | ||
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/-! | ||
# Topology generated by its restrictions to subsets | ||
We say that restrictions of the topology on `X` to sets from a family `S` | ||
generates the original topology, | ||
if either of the following equivalent conditions hold: | ||
- a set which is relatively open in each `s ∈ S` is open; | ||
- a set which is relatively closed in each `s ∈ S` is closed; | ||
- for any topological space `Y`, a function `f : X → Y` is continuous | ||
provided that it is continuous on each `s ∈ S`. | ||
We use the first condition as the definition | ||
(see `RestrictGenTopology` in `Topology/Defs/Induced.lean`), | ||
and provide the others as corollaries. | ||
## Main results | ||
- `RestrictGenTopology.of_seq`: if `X` is a sequential space | ||
and `S` contains all sets of the form `insert x (Set.range u)`, | ||
where `u : ℕ → X` is a sequence that converges to `x`, | ||
then we have `RestrictGenTopology S`; | ||
- `RestrictGenTopology.isCompact_of_seq`: specialization of the previous lemma | ||
to the case `S = {K | IsCompact K}`. | ||
-/ | ||
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open Set Filter | ||
open scoped Topology | ||
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variable {X : Type*} [TopologicalSpace X] {S : Set (Set X)} {t : Set X} {x : X} | ||
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namespace RestrictGenTopology | ||
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protected theorem isOpen_iff (hS : RestrictGenTopology S) : | ||
IsOpen t ↔ ∀ s ∈ S, IsOpen ((↑) ⁻¹' t : Set s) := | ||
⟨fun ht _ _ ↦ ht.preimage continuous_subtype_val, hS.1 t⟩ | ||
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protected theorem isClosed_iff (hS : RestrictGenTopology S) : | ||
IsClosed t ↔ ∀ s ∈ S, IsClosed ((↑) ⁻¹' t : Set s) := by | ||
simp only [← isOpen_compl_iff, hS.isOpen_iff, preimage_compl] | ||
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protected theorem continuous_iff {Y : Type*} [TopologicalSpace Y] {f : X → Y} | ||
(hS : RestrictGenTopology S) : | ||
Continuous f ↔ ∀ s ∈ S, ContinuousOn f s := | ||
⟨fun h _ _ ↦ h.continuousOn, fun h ↦ continuous_def.2 fun _u hu ↦ hS.isOpen_iff.2 fun s hs ↦ | ||
hu.preimage <| (h s hs).restrict⟩ | ||
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theorem of_continuous_prop (h : ∀ f : X → Prop, (∀ s ∈ S, ContinuousOn f s) → Continuous f) : | ||
RestrictGenTopology S where | ||
isOpen_of_forall_induced u hu := by | ||
simp only [continuousOn_iff_continuous_restrict, continuous_Prop] at * | ||
exact h _ hu | ||
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theorem of_isClosed (h : ∀ t : Set X, (∀ s ∈ S, IsClosed ((↑) ⁻¹' t : Set s)) → IsClosed t) : | ||
RestrictGenTopology S := | ||
⟨fun _t ht ↦ isClosed_compl_iff.1 <| h _ fun s hs ↦ (ht s hs).isClosed_compl⟩ | ||
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protected theorem enlarge {T} (hS : RestrictGenTopology S) (hT : ∀ s ∈ S, ∃ t ∈ T, s ⊆ t) : | ||
RestrictGenTopology T := | ||
of_continuous_prop fun _f hf ↦ hS.continuous_iff.2 fun s hs ↦ | ||
let ⟨t, htT, hst⟩ := hT s hs; (hf t htT).mono hst | ||
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protected theorem mono {T} (hS : RestrictGenTopology S) (hT : S ⊆ T) : RestrictGenTopology T := | ||
hS.enlarge fun s hs ↦ ⟨s, hT hs, Subset.rfl⟩ | ||
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/-- If `X` is a sequential space | ||
and `S` contains each set of the form `insert x (Set.range u)` | ||
where `u : ℕ → X` is a sequence and `x` is its limit, | ||
then topology on `X` is generated by its restrictions to the sets of `S`. -/ | ||
lemma of_seq [SequentialSpace X] | ||
(h : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S) : | ||
RestrictGenTopology S := by | ||
refine of_isClosed fun t ht ↦ IsSeqClosed.isClosed fun u x hut hux ↦ ?_ | ||
rcases isClosed_induced_iff.1 (ht _ (h hux)) with ⟨s, hsc, hst⟩ | ||
rw [Subtype.preimage_val_eq_preimage_val_iff, Set.ext_iff] at hst | ||
suffices x ∈ s by specialize hst x; simp_all | ||
refine hsc.mem_of_tendsto hux <| eventually_of_forall fun k ↦ ?_ | ||
specialize hst (u k) | ||
simp_all | ||
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/-- A sequential space is compactly generated. -/ | ||
lemma isCompact_of_seq [SequentialSpace X] : RestrictGenTopology {K : Set X | IsCompact K} := | ||
of_seq fun _u _x hux ↦ hux.isCompact_insert_range | ||
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end RestrictGenTopology |
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