Theorems · Theorem · general topology
TopologicalSpace.Closeds.coe_eq_singleton_of_isAtom
∀ {α : Type u_2} [inst : TopologicalSpace α] [T0Space α] {s : TopologicalSpace.Closeds α}, IsAtom s → ∃ a, ↑s = {a}- Defined in
- Mathlib.Topology.Sets.Closeds
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceT0Space
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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
- SetLike.coestatement and proof · cited by 8,199
- Set.Nonemptyproof · cited by 2,627
- IsClosedproof · cited by 1,639
- SetLike.coe_injectiveproof · cited by 374
- T0Spacestatement and proof · cited by 179
- TopologicalSpace.Closedsstatement and proof · cited by 168
- IsAtomstatement and proof · cited by 130
- IsAtom.le_iff_eqproof · cited by 8
- TopologicalSpace.Closeds.isClosed'proof · cited by 5
- minimal_nonempty_closed_eq_singletonproof · cited by 2
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