Theorems · Theorem · general topology
closure_compl_singleton
∀ {X : Type u} [inst : TopologicalSpace X] (x : X) [(nhdsWithin x {x}ᶜ).NeBot], closure {x}ᶜ = Set.univIf x is not an isolated point of a topological space, then the closure of {x}ᶜ is the whole
space.
- Defined in
- Mathlib.Topology.ClusterPt
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceFilter.NeBot
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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
- Set.univstatement · cited by 3,945
- Compl.complstatement and proof · cited by 2,925
- nhdsWithinstatement and proof · cited by 1,912
- closurestatement · cited by 1,254
- Filter.NeBotstatement and proof · cited by 853
- Dense.closure_eqproof · cited by 24
- dense_compl_singletonproof · cited by 6
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