File: Mathlib/Topology/LocallyClosed.lean

full_name: subset_coborder
code: lemma subset_coborder :
    s ⊆ coborder s 
goal: s ⊆ coborder s

full_name: coborder_inter_closure
code: lemma coborder_inter_closure :
    coborder s ∩ closure s = s 
goal: coborder s ∩ closure s = s

full_name: closure_inter_coborder
code: lemma closure_inter_coborder :
    closure s ∩ coborder s = s 
goal: closure s ∩ coborder s = s

full_name: coborder_eq_union_frontier_compl
code: lemma coborder_eq_union_frontier_compl :
    coborder s = s ∪ (frontier s)ᶜ 
goal: coborder s = s ∪ (frontier s)ᶜ

full_name: coborder_eq_univ_iff
code: lemma coborder_eq_univ_iff :
    coborder s = univ ↔ IsClosed s 
goal: coborder s = Set.univ ↔ IsClosed s

full_name: coborder_eq_compl_frontier_iff
code: lemma coborder_eq_compl_frontier_iff :
    coborder s = (frontier s)ᶜ ↔ IsOpen s 
goal: coborder s = (frontier s)ᶜ ↔ IsOpen s

full_name: IsOpenMap.coborder_preimage_subset
code: lemma IsOpenMap.coborder_preimage_subset (hf : IsOpenMap f) (s : Set Y) :
    coborder (f ⁻¹' s) ⊆ f ⁻¹' (coborder s) 
goal: coborder (f ⁻¹' s) ⊆ f ⁻¹' coborder s

full_name: Continuous.preimage_coborder_subset
code: lemma Continuous.preimage_coborder_subset (hf : Continuous f) (s : Set Y) :
    f ⁻¹' (coborder s) ⊆ coborder (f ⁻¹' s) 
goal: f ⁻¹' coborder s ⊆ coborder (f ⁻¹' s)

full_name: coborder_preimage
code: lemma coborder_preimage (hf : IsOpenMap f) (hf' : Continuous f) (s : Set Y) :
    coborder (f ⁻¹' s) = f ⁻¹' (coborder s) 
goal: coborder (f ⁻¹' s) = f ⁻¹' coborder s

full_name: OpenEmbedding.coborder_preimage
code: protected
lemma OpenEmbedding.coborder_preimage (hf : OpenEmbedding f) (s : Set Y) :
    coborder (f ⁻¹' s) = f ⁻¹' (coborder s) 
goal: coborder (f ⁻¹' s) = f ⁻¹' coborder s

full_name: isClosed_preimage_val_coborder
code: lemma isClosed_preimage_val_coborder :
    IsClosed (coborder s ↓∩ s) 
goal: IsClosed (coborder s ↓∩ s)

full_name: IsOpen.isLocallyClosed
code: lemma IsOpen.isLocallyClosed (hs : IsOpen s) : IsLocallyClosed s 
goal: IsLocallyClosed s

full_name: IsClosed.isLocallyClosed
code: lemma IsClosed.isLocallyClosed (hs : IsClosed s) : IsLocallyClosed s 
goal: IsLocallyClosed s

full_name: IsLocallyClosed.inter
code: lemma IsLocallyClosed.inter (hs : IsLocallyClosed s) (ht : IsLocallyClosed t) :
    IsLocallyClosed (s ∩ t) 
goal: IsLocallyClosed (s ∩ t)

full_name: IsLocallyClosed.preimage
code: lemma IsLocallyClosed.preimage {s : Set Y} (hs : IsLocallyClosed s)
    {f : X → Y} (hf : Continuous f) :
    IsLocallyClosed (f ⁻¹' s) 
goal: IsLocallyClosed (f ⁻¹' s)

full_name: Inducing.isLocallyClosed_iff
code: nonrec
lemma Inducing.isLocallyClosed_iff {s : Set X}
    {f : X → Y} (hf : Inducing f) :
    IsLocallyClosed s ↔ ∃ s' : Set Y, IsLocallyClosed s' ∧ f ⁻¹' s' = s 
goal: IsLocallyClosed s ↔ ∃ s', IsLocallyClosed s' ∧ f ⁻¹' s' = s

full_name: Embedding.isLocallyClosed_iff
code: lemma Embedding.isLocallyClosed_iff {s : Set X}
    {f : X → Y} (hf : Embedding f) :
    IsLocallyClosed s ↔ ∃ s' : Set Y, IsLocallyClosed s' ∧ s' ∩ range f = f '' s 
goal: IsLocallyClosed s ↔ ∃ s', IsLocallyClosed s' ∧ s' ∩ Set.range f = f '' s

full_name: IsLocallyClosed.image
code: lemma IsLocallyClosed.image {s : Set X} (hs : IsLocallyClosed s)
    {f : X → Y} (hf : Inducing f) (hf' : IsLocallyClosed (range f)) :
    IsLocallyClosed (f '' s) 
goal: IsLocallyClosed (f '' s)

full_name: isLocallyClosed_tfae
code: /--
A set `s` is locally closed if one of the equivalent conditions below hold
1. It is the intersection of some open set and some closed set.
2. The coborder `(closure s \ s)ᶜ` is open.
3. `s` is closed in some neighborhood of `x` for all `x ∈ s`.
4. Every `x ∈ s` has some open neighborhood `U` such that `U ∩ closure s ⊆ s`.
5. `s` is open in the closure of `s`.
-/
lemma isLocallyClosed_tfae (s : Set X) :
    List.TFAE
    [ IsLocallyClosed s,
      IsOpen (coborder s),
      ∀ x ∈ s, ∃ U ∈ 𝓝 x, IsClosed (U ↓∩ s),
      ∀ x ∈ s, ∃ U, x ∈ U ∧ IsOpen U ∧ U ∩ closure s ⊆ s,
      IsOpen (closure s ↓∩ s)] 
goal: [IsLocallyClosed s, IsOpen (coborder s), ∀ x ∈ s, ∃ U ∈ 𝓝 x, IsClosed (U ↓∩ s),
    ∀ x ∈ s, ∃ U, x ∈ U ∧ IsOpen U ∧ U ∩ closure s ⊆ s, IsOpen (closure s ↓∩ s)].TFAE

full_name: isLocallyClosed_iff_isOpen_coborder
code: lemma isLocallyClosed_iff_isOpen_coborder : IsLocallyClosed s ↔ IsOpen (coborder s) 
goal: IsLocallyClosed s ↔ IsOpen (coborder s)

full_name: IsLocallyClosed.isOpen_preimage_val_closure
code: lemma IsLocallyClosed.isOpen_preimage_val_closure (hs : IsLocallyClosed s) :
    IsOpen (closure s ↓∩ s) 
goal: IsOpen (closure s ↓∩ s)