Theorems · Theorem · general topology
closure_singleton
∀ {X : Type u_1} [inst : TopologicalSpace X] [T1Space X] {x : X}, closure {x} = {x}- Defined in
- Mathlib.Topology.Separation.Basic
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceT1Space
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- closurestatement · cited by 1,254
- T1Spacestatement and proof · cited by 275
- IsClosed.closure_eqproof · cited by 139
- isClosed_singletonproof · cited by 66
Cited by8
Results whose statement or proof uses this declaration.
- ContinuousWithinAt.eq_const_of_mem_closureproof · cited by 1
- closure_smul₀proof · cited by 1
- denseRange_zpow_iff_surjectiveproof · cited by 1
- denseRange_zsmul_iff_surjectiveproof · cited by 1
- TopologicalSpace.Closeds.isAtom_coeproof · cited by 0
- UniformSpace.Completion.mul_hatInv_cancelproof · cited by 0