Theorems · Theorem · general topology
dense_compl_singleton
∀ {X : Type u} [inst : TopologicalSpace X] (x : X) [(nhdsWithin x {x}ᶜ).NeBot], Dense {x}ᶜIf x is not an isolated point of a topological space, then {x}ᶜ is dense in the whole
space.
- Defined in
- Mathlib.Topology.ClusterPt
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceFilter.NeBot
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 · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Compl.complstatement and proof · cited by 2,925
- nhdsWithinstatement and proof · cited by 1,912
- eq_or_neproof · cited by 1,117
- Filter.NeBotstatement and proof · cited by 853
- Densestatement · cited by 359
- subset_closureproof · cited by 309
- mem_closure_iff_nhdsWithin_neBotproof · cited by 16
Cited by6
Results whose statement or proof uses this declaration.
- Dense.sdiff_singletonproof · cited by 2
- interior_singletonproof · cited by 2
- DirichletCharacter.LFunction_changeLevelproof · cited by 1
- OnePoint.denseRange_coeproof · cited by 1
- not_isOpen_singletonproof · cited by 0
- closure_compl_singletonproof · cited by 0