Theorems · Theorem · general topology
isDiscrete_iff_nhdsNE
∀ {Y : Type u_2} [inst : TopologicalSpace Y] {S : Set Y},
IsDiscrete S ↔ ∀ x ∈ S, nhdsWithin x {x}ᶜ ⊓ Filter.principal S = ⊥- Defined in
- Mathlib.Topology.DiscreteSubset
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- Bot.botstatement and proof · cited by 4,720
- Compl.complstatement and proof · cited by 2,925
- nhdsWithinstatement and proof · cited by 1,912
- Filter.principalstatement and proof · cited by 740
- IsDiscretestatement · cited by 86
- isDiscrete_iff_discreteTopologyproof · cited by 6
- discreteTopology_subtype_iffproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- isDiscrete_of_codiscreteWithinproof · cited by 3
- isClosed_and_discrete_iffproof · cited by 2