Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
/- | |
Copyright (c) 2022 Floris van Doorn. All rights reserved. | |
Released under Apache 2.0 license as described in the file LICENSE. | |
Authors: Floris van Doorn, Patrick Massot | |
-/ | |
import topology.basic | |
/-! | |
# Neighborhoods of a set | |
In this file we define the filter `πΛ’ s` or `nhds_set s` consisting of all neighborhoods of a set | |
`s`. | |
## Main Properties | |
There are a couple different notions equivalent to `s β πΛ’ t`: | |
* `s β interior t` using `subset_interior_iff_mem_nhds_set` | |
* `β (x : Ξ±), x β t β s β π x` using `mem_nhds_set_iff_forall` | |
* `β U : set Ξ±, is_open U β§ t β U β§ U β s` using `mem_nhds_set_iff_exists` | |
Furthermore, we have the following results: | |
* `monotone_nhds_set`: `πΛ’` is monotone | |
* In Tβ-spaces, `πΛ’`is strictly monotone and hence injective: | |
`strict_mono_nhds_set`/`injective_nhds_set`. These results are in `topology.separation`. | |
-/ | |
open set filter | |
open_locale topological_space | |
variables {Ξ± Ξ² : Type*} [topological_space Ξ±] [topological_space Ξ²] | |
{s t sβ sβ tβ tβ : set Ξ±} {x : Ξ±} | |
/-- The filter of neighborhoods of a set in a topological space. -/ | |
def nhds_set (s : set Ξ±) : filter Ξ± := | |
Sup (nhds '' s) | |
localized "notation `πΛ’` := nhds_set" in topological_space | |
lemma mem_nhds_set_iff_forall : s β πΛ’ t β β (x : Ξ±), x β t β s β π x := | |
by simp_rw [nhds_set, filter.mem_Sup, ball_image_iff] | |
lemma subset_interior_iff_mem_nhds_set : s β interior t β t β πΛ’ s := | |
by simp_rw [mem_nhds_set_iff_forall, subset_interior_iff_nhds] | |
lemma mem_nhds_set_iff_exists : s β πΛ’ t β β U : set Ξ±, is_open U β§ t β U β§ U β s := | |
by { rw [β subset_interior_iff_mem_nhds_set, subset_interior_iff] } | |
lemma has_basis_nhds_set (s : set Ξ±) : (πΛ’ s).has_basis (Ξ» U, is_open U β§ s β U) (Ξ» U, U) := | |
β¨Ξ» t, by simp [mem_nhds_set_iff_exists, and_assoc]β© | |
lemma is_open.mem_nhds_set (hU : is_open s) : s β πΛ’ t β t β s := | |
by rw [β subset_interior_iff_mem_nhds_set, interior_eq_iff_open.mpr hU] | |
@[simp] lemma nhds_set_singleton : πΛ’ {x} = π x := | |
by { ext, | |
rw [β subset_interior_iff_mem_nhds_set, β mem_interior_iff_mem_nhds, singleton_subset_iff] } | |
lemma mem_nhds_set_interior : s β πΛ’ (interior s) := | |
subset_interior_iff_mem_nhds_set.mp subset.rfl | |
lemma mem_nhds_set_empty : s β πΛ’ (β : set Ξ±) := | |
subset_interior_iff_mem_nhds_set.mp $ empty_subset _ | |
@[simp] lemma nhds_set_empty : πΛ’ (β : set Ξ±) = β₯ := | |
by { ext, simp [mem_nhds_set_empty] } | |
@[simp] lemma nhds_set_univ : πΛ’ (univ : set Ξ±) = β€ := | |
by { ext, rw [β subset_interior_iff_mem_nhds_set, univ_subset_iff, interior_eq_univ, mem_top] } | |
lemma monotone_nhds_set : monotone (πΛ’ : set Ξ± β filter Ξ±) := | |
Ξ» s t hst, Sup_le_Sup $ image_subset _ hst | |
@[simp] lemma nhds_set_union (s t : set Ξ±) : πΛ’ (s βͺ t) = πΛ’ s β πΛ’ t := | |
by simp only [nhds_set, image_union, Sup_union] | |
lemma union_mem_nhds_set (hβ : sβ β πΛ’ tβ) (hβ : sβ β πΛ’ tβ) : sβ βͺ sβ β πΛ’ (tβ βͺ tβ) := | |
by { rw nhds_set_union, exact union_mem_sup hβ hβ } | |
/-- Preimage of a set neighborhood of `t` under a continuous map `f` is a set neighborhood of `s` | |
provided that `f` maps `s` to `t`. -/ | |
lemma continuous.tendsto_nhds_set {f : Ξ± β Ξ²} {t : set Ξ²} (hf : continuous f) | |
(hst : maps_to f s t) : tendsto f (πΛ’ s) (πΛ’ t) := | |
((has_basis_nhds_set s).tendsto_iff (has_basis_nhds_set t)).mpr $ Ξ» U hU, | |
β¨f β»ΒΉ' U, β¨hU.1.preimage hf, hst.mono subset.rfl hU.2β©, Ξ» x, idβ© | |