Theorems · Theorem · general topology
IsOpenMap.preimage_interior_eq_interior_preimage
∀ {X : Type u_1} {Y : Type u_2} {f : X → Y} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y],
IsOpenMap f → Continuous f → ∀ (s : Set Y), f ⁻¹' interior s = interior (f ⁻¹' s)- Defined in
- Mathlib.Topology.Maps.Basic
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.preimagestatement · cited by 4,946
- Continuousstatement and proof · cited by 2,592
- interiorstatement · cited by 714
- IsOpenMapstatement and proof · cited by 253
- Set.Subset.antisymmproof · cited by 213
- preimage_interior_subset_interior_preimageproof · cited by 14
- IsOpenMap.interior_preimage_subset_preimage_interiorproof · cited by 2
Cited by8
Results whose statement or proof uses this declaration.
- Complex.interior_preimage_improof · cited by 3
- Complex.interior_preimage_reproof · cited by 3
- interior_halfSpaceproof · cited by 2
- Homeomorph.preimage_interiorproof · cited by 2
- AffineBasis.interior_convexHullproof · cited by 1
- SeparationQuotient.preimage_mk_interiorproof · cited by 0
- ContinuousLinearMap.interior_preimageproof · cited by 0
- interior_euclideanQuadrantproof · cited by 0