diff --git a/Mathlib/Topology/Basic.lean b/Mathlib/Topology/Basic.lean
index f08a0fafc0a2c..1d353cdafed19 100644
--- a/Mathlib/Topology/Basic.lean
+++ b/Mathlib/Topology/Basic.lean
@@ -1721,9 +1721,8 @@ theorem closure_image_closure (h : Continuous f) :
     (closure_mono <| image_subset _ subset_closure)
 
 theorem closure_subset_preimage_closure_image (h : Continuous f) :
-    closure s ⊆ f ⁻¹' closure (f '' s) := by
-  rw [← Set.image_subset_iff]
-  exact image_closure_subset_closure_image h
+    closure s ⊆ f ⁻¹' closure (f '' s) :=
+  (mapsTo_image _ _).closure h
 #align closure_subset_preimage_closure_image closure_subset_preimage_closure_image
 
 theorem map_mem_closure {t : Set Y} (hf : Continuous f)
@@ -1737,6 +1736,15 @@ theorem Set.MapsTo.closure_left {t : Set Y} (h : MapsTo f s t)
   ht.closure_eq ▸ h.closure hc
 #align set.maps_to.closure_left Set.MapsTo.closure_left
 
+theorem Filter.Tendsto.lift'_closure (hf : Continuous f) {l l'} (h : Tendsto f l l') :
+    Tendsto f (l.lift' closure) (l'.lift' closure) :=
+  tendsto_lift'.2 fun s hs ↦ by
+    filter_upwards [mem_lift' (h hs)] using (mapsTo_preimage _ _).closure hf
+
+theorem tendsto_lift'_closure_nhds (hf : Continuous f) (x : X) :
+    Tendsto f ((𝓝 x).lift' closure) ((𝓝 (f x)).lift' closure) :=
+  (hf.tendsto x).lift'_closure hf
+
 /-!
 ### Function with dense range
 -/
diff --git a/Mathlib/Topology/Separation.lean b/Mathlib/Topology/Separation.lean
index a2ec8140b62dd..f4f7a20f636d9 100644
--- a/Mathlib/Topology/Separation.lean
+++ b/Mathlib/Topology/Separation.lean
@@ -1079,10 +1079,13 @@ theorem IsCompact.finite_compact_cover {s : Set X} (hs : IsCompact s) {ι : Type
   · simp only [Finset.set_biUnion_insert_update _ hx, hK, h3K]
 #align is_compact.finite_compact_cover IsCompact.finite_compact_cover
 
-theorem Inducing.r1Space [TopologicalSpace Y] {f : Y → X} (hf : Inducing f) : R1Space Y where
-  specializes_or_disjoint_nhds _ _ := by
-    simpa only [← hf.specializes_iff, hf.nhds_eq_comap, or_iff_not_imp_left,
-      ← disjoint_nhds_nhds_iff_not_specializes] using Filter.disjoint_comap
+theorem R1Space.of_continuous_specializes_imp [TopologicalSpace Y] {f : Y → X} (hc : Continuous f)
+    (hspec : ∀ x y, f x ⤳ f y → x ⤳ y) : R1Space Y where
+  specializes_or_disjoint_nhds x y := (specializes_or_disjoint_nhds (f x) (f y)).imp (hspec x y) <|
+    ((hc.tendsto _).disjoint · (hc.tendsto _))
+
+theorem Inducing.r1Space [TopologicalSpace Y] {f : Y → X} (hf : Inducing f) : R1Space Y :=
+  .of_continuous_specializes_imp hf.continuous fun _ _ ↦ hf.specializes_iff.1
 
 protected theorem R1Space.induced (f : Y → X) : @R1Space Y (.induced f ‹_›) :=
   @Inducing.r1Space _ _ _ _ (.induced f _) f (inducing_induced f)
@@ -1396,6 +1399,14 @@ theorem exists_open_nhds_disjoint_closure [T25Space X] {x y : X} (h : x ≠ y) :
       (disjoint_lift'_closure_nhds.2 h)
 #align exists_open_nhds_disjoint_closure exists_open_nhds_disjoint_closure
 
+theorem T25Space.of_injective_continuous [TopologicalSpace Y] [T25Space Y] {f : X → Y}
+    (hinj : Injective f) (hcont : Continuous f) : T25Space X where
+  t2_5 x y hne := (tendsto_lift'_closure_nhds hcont x).disjoint (t2_5 <| hinj.ne hne)
+    (tendsto_lift'_closure_nhds hcont y)
+
+instance [T25Space X] {p : X → Prop} : T25Space {x // p x} :=
+  .of_injective_continuous Subtype.val_injective continuous_subtype_val
+
 section limUnder
 
 variable [T2Space X] {f : Filter X}