Theorems · Theorem · general topology
infinite_of_mem_nhds
∀ {X : Type u_3} [inst : TopologicalSpace X] [T1Space X] (x : X) [hx : (nhdsWithin x {x}ᶜ).NeBot] {s : Set X},
s ∈ nhds x → s.InfiniteIf the punctured neighborhoods of a point form a nontrivial filter, then any neighborhood is infinite.
- Defined in
- Mathlib.Topology.Separation.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites13
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
- Filterstatement · cited by 8,121
- nhdsstatement and proof · cited by 5,554
- Compl.complstatement and proof · cited by 2,925
- nhdsWithinstatement and proof · cited by 1,912
- Set.Finiteproof · cited by 1,814
- Filter.NeBotstatement and proof · cited by 853
- T1Spacestatement and proof · cited by 275
- Set.Infinitestatement · cited by 263
- Filter.NeBot.ne'proof · cited by 7
- isOpen_singleton_of_finite_mem_nhdsproof · cited by 2
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