feat: three tiny homeomorph lemmas (#9166)

Mathlib/Topology/Algebra/Group/Basic.lean

@@ -334,6 +334,10 @@ protected def Homeomorph.inv (G : Type*) [TopologicalSpace G] [InvolutiveInv G]
 #align homeomorph.inv Homeomorph.inv
 #align homeomorph.neg Homeomorph.neg
 
+@[to_additive (attr := simp)]
+lemma Homeomorph.coe_inv {G : Type*} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] :
+    ⇑(Homeomorph.inv G) = Inv.inv := rfl
+
 @[to_additive]
 theorem isOpenMap_inv : IsOpenMap (Inv.inv : G → G) :=
   (Homeomorph.inv _).isOpenMap

Mathlib/Topology/Homeomorph.lean

@@ -234,6 +234,9 @@ theorem preimage_image (h : X ≃ₜ Y) (s : Set X) : h ⁻¹' (h '' s) = s :=
   h.toEquiv.preimage_image s
 #align homeomorph.preimage_image Homeomorph.preimage_image
 
+lemma image_compl (h : X ≃ₜ Y) (s : Set X) : h '' (sᶜ) = (h '' s)ᶜ :=
+  h.toEquiv.image_compl s
+
 protected theorem inducing (h : X ≃ₜ Y) : Inducing h :=
   inducing_of_inducing_compose h.continuous h.symm.continuous <| by
     simp only [symm_comp_self, inducing_id]
@@ -427,6 +430,11 @@ theorem map_nhds_eq (h : X ≃ₜ Y) (x : X) : map h (𝓝 x) = 𝓝 (h x) :=
   h.embedding.map_nhds_of_mem _ (by simp)
 #align homeomorph.map_nhds_eq Homeomorph.map_nhds_eq
 
+@[simp]
+theorem map_punctured_nhds_eq (h : X ≃ₜ Y) (x : X) : map h (𝓝[≠] x) = 𝓝[≠] (h x) := by
+  convert h.embedding.map_nhdsWithin_eq ({x}ᶜ) x
+  rw [h.image_compl, Set.image_singleton]
+
 theorem symm_map_nhds_eq (h : X ≃ₜ Y) (x : X) : map h.symm (𝓝 (h x)) = 𝓝 x := by
   rw [h.symm.map_nhds_eq, h.symm_apply_apply]
 #align homeomorph.symm_map_nhds_eq Homeomorph.symm_map_nhds_eq