Theorems · Theorem · functional analysis
Submodule.dense_iff_topologicalClosure_eq_top
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : TopologicalSpace M] [inst_2 : AddCommMonoid M]
[inst_3 : Module R M] [inst_4 : ContinuousConstSMul R M] [inst_5 : ContinuousAdd M] {s : Submodule R M},
Dense ↑s ↔ s.topologicalClosure = ⊤A subspace is dense iff its topological closure is the entire space.
- Defined in
- Mathlib.Topology.Algebra.Module.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- TopologicalSpacestatement and proof · cited by 24,529
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Top.topstatement and proof · cited by 9,680
- SetLike.coestatement and proof · cited by 8,199
- Submodulestatement and proof · cited by 7,192
- Set.univproof · cited by 3,945
- closureproof · cited by 1,254
- ContinuousConstSMulstatement and proof · cited by 832
- ContinuousAddstatement and proof · cited by 777
- Densestatement and proof · cited by 359
Cited by1
Results whose statement or proof uses this declaration.
- Submodule.isClosed_or_dense_of_isCoatomproof · cited by 1