Theorems · Theorem · general topology
dense_iff_closure_eq
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, Dense s ↔ closure s = Set.univ- Defined in
- Mathlib.Topology.Closure
- Cited by
- 26 results in Mathlib
- Foundations
- Depth 23 from the axioms · uses propext, Quot.sound
- Assumes
- TopologicalSpace
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.univstatement · cited by 3,945
- closurestatement · cited by 1,254
- Densestatement · cited by 359
- Set.eq_univ_iff_forallproof · cited by 93
Cited by26
Results whose statement or proof uses this declaration.
- Dense.closure_eqproof · cited by 24
- denseRange_iff_closure_rangeproof · cited by 5
- interior_eq_empty_iff_dense_complproof · cited by 5
- aeconst_of_dense_setOfPred_preimage_smul_aeproof · cited by 4
- aeconst_of_dense_setOfPred_preimage_vadd_aeproof · cited by 4
- IsDenseEmbedding.subtypeproof · cited by 3
- ContinuousMapZero.adjoin_id_denseproof · cited by 3
- stronglyMeasurable_derivWithin_Iciproof · cited by 2
- denseRange_zsmul_iff_surjectiveproof · cited by 1
- range_deriv_subset_closure_span_imageproof · cited by 1
- AddSubgroup.discrete_iff_addCyclicproof · cited by 1
- dense_coborderproof · cited by 1