Theorems · Theorem · general topology
set_smul_mem_nhds_zero_iff
∀ {α : Type u_2} {G₀ : Type u_6} [inst : GroupWithZero G₀] [inst_1 : AddMonoid α] [inst_2 : DistribMulAction G₀ α]
[inst_3 : TopologicalSpace α] [ContinuousConstSMul G₀ α] {s : Set α} {c : G₀}, c ≠ 0 → (c • s ∈ nhds 0 ↔ s ∈ nhds 0)- Defined in
- Mathlib.Topology.Algebra.ConstMulAction
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- AddMonoidstatement and proof · cited by 2,864
- ContinuousConstSMulstatement and proof · cited by 832
- GroupWithZerostatement and proof · cited by 691
- smul_zeroproof · cited by 665
- Set.smulSetstatement · cited by 608
- DistribMulActionstatement and proof · cited by 584
- smul_mem_nhds_smul_iff₀proof · cited by 2
Cited by10
Results whose statement or proof uses this declaration.
- Asymptotics.isLittleOTVS_oneproof · cited by 4
- balancedCore_mem_nhds_zeroproof · cited by 4
- Seminorm.continuousAt_zero'proof · cited by 3
- FiniteDimensional.of_totallyBounded_nhds_zeroproof · cited by 3
- continuousAt_gaugeproof · cited by 2
- ContinuousSMul.topology_eq_of_nhds_inf_principal_eqproof · cited by 1
- tendsto_gauge_nhds_zero_nhdsGEproof · cited by 1
- UniformConvergenceCLM.continuous_of_continuous_uncurryproof · cited by 1
- Asymptotics.IsBigOTVS.of_egauge_le_mulproof · cited by 0
- IsCompactOperator.continuousproof · cited by 0