Theorems · Theorem · general topology
IsOpenMap.preimage_frontier_subset_frontier_preimage
∀ {X : Type u_1} {Y : Type u_2} {f : X → Y} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y],
IsOpenMap f → ∀ {s : Set Y}, f ⁻¹' frontier s ⊆ frontier (f ⁻¹' s)- Defined in
- Mathlib.Topology.Maps.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites8
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 and proof · cited by 4,946
- IsOpenMapstatement and proof · cited by 253
- frontierstatement · cited by 214
- Set.inter_subset_interproof · cited by 66
- frontier_eq_closure_inter_closureproof · cited by 12
- IsOpenMap.preimage_closure_subset_closure_preimageproof · cited by 5
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