Theorems · Theorem · general topology
Dense.preimage
∀ {X : Type u_1} {Y : Type u_2} {f : X → Y} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {s : Set Y},
Dense s → IsOpenMap f → Dense (f ⁻¹' s)Preimage of a dense set under an open map is dense.
- Defined in
- Mathlib.Topology.Maps.Basic
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- Densestatement and proof · cited by 359
- IsOpenMapstatement and proof · cited by 253
- IsOpenMap.preimage_closure_subset_closure_preimageproof · cited by 5
Cited by8
Results whose statement or proof uses this declaration.
- tendsto_residual_of_isOpenMapproof · cited by 3
- IsOpenQuotientMap.dense_preimage_iffproof · cited by 3
- IsOpenMap.separableSpace_of_injectiveproof · cited by 1
- aeconst_of_dense_aestabilizer_smulproof · cited by 1
- aeconst_of_dense_aestabilizer_vaddproof · cited by 1
- AlgebraicGeometry.IsDominant.of_comp_of_isOpenImmersionproof · cited by 1
- TopologicalSpace.NonemptyCompacts.dense_setOfPred_finiteproof · cited by 1
- Topology.IsOpenEmbedding.baireSpaceproof · cited by 1