Theorems · Theorem · functional analysis
IsClosed.submodule_topologicalClosure_eq
∀ {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},
IsClosed ↑s → s.topologicalClosure = sThe topological closure of a closed submodule s is equal to s.
- Defined in
- Mathlib.Topology.Algebra.Module.Basic
- Cited by
- 2 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.
Cites12
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
- SetLike.coestatement and proof · cited by 8,199
- Submodulestatement and proof · cited by 7,192
- IsClosedstatement and proof · cited by 1,639
- ContinuousConstSMulstatement and proof · cited by 832
- ContinuousAddstatement and proof · cited by 777
- IsClosed.closure_eqproof · cited by 139
- SetLike.ext'proof · cited by 60
- Submodule.topologicalClosurestatement · cited by 50
Cited by2
Results whose statement or proof uses this declaration.
- LinearPMap.IsClosed.isClosableproof · cited by 2
- Submodule.topologicalClosure_eq_selfproof · cited by 2