Theorems · Theorem · functional analysis
Units.isOpen
∀ {R : Type u_1} [inst : NormedRing R] [HasSummableGeomSeries R], IsOpen {x | IsUnit x}The group of units of a normed ring with summable geometric series is an open subset of the ring.
- Defined in
- Mathlib.Analysis.Normed.Ring.Units
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Set.ofPredstatement and proof · cited by 6,101
- Norm.normproof · cited by 5,413
- Set.univproof · cited by 3,945
- Unitsproof · cited by 2,804
- Nontrivialproof · cited by 2,416
- IsOpenstatement and proof · cited by 2,400
- Units.valproof · cited by 1,966
- IsUnitstatement and proof · cited by 1,602
- NormedRingstatement and proof · cited by 924
- Metric.ballproof · cited by 735
- inv_posproof · cited by 124
- Units.isUnitproof · cited by 116
Cited by6
Results whose statement or proof uses this declaration.
- nonunits.isClosedproof · cited by 2
- Units.isOpenEmbedding_valproof · cited by 2
- spectrum.isOpen_resolventSetproof · cited by 2
- contDiffAt_ringInverseproof · cited by 2
- ContinuousLinearEquiv.isOpenproof · cited by 1
- Units.nhdsproof · cited by 0