Theorems · Theorem · general topology
interior_singleton
∀ {X : Type u} [inst : TopologicalSpace X] (x : X) [(nhdsWithin x {x}ᶜ).NeBot], interior {x} = ∅If x is not an isolated point of a topological space, then the interior of {x} is empty.
- Defined in
- Mathlib.Topology.ClusterPt
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 76 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.
Cites8
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
- Filter.NeBotstatement and proof · cited by 853
- interiorstatement · cited by 714
- dense_compl_singletonproof · cited by 6
- interior_eq_empty_iff_dense_complproof · cited by 5
Cited by2
Results whose statement or proof uses this declaration.
- interior_closedBall'proof · cited by 3
- Convex.nontrivial_iff_nonempty_interiorproof · cited by 2