Theorems · Theorem · functional analysis
Submodule.IsTopCompl.isClosed
∀ {R : Type u_1} [inst : Ring R] {M : Type u_2} [inst_1 : TopologicalSpace M] [inst_2 : AddCommGroup M]
[inst_3 : Module R M] {p q : Submodule R M} [T1Space ↥q] [ContinuousSub M], Submodule.IsTopCompl p q → IsClosed ↑pIf p and q are topological complements and q is Hausdorff, then p is closed.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 80 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
- AddCommGroupstatement and proof · cited by 12,871
- SetLike.coestatement · cited by 8,199
- Ringstatement and proof · cited by 7,463
- Submodulestatement and proof · cited by 7,192
- IsClosedstatement · cited by 1,639
- T1Spacestatement and proof · cited by 275
- Submodule.IsTopComplstatement and proof · cited by 89
- ContinuousSubstatement and proof · cited by 48
- Submodule.IsTopCompl.symmproof · cited by 14
- Submodule.IsTopCompl.isClosed'proof · cited by 4
Cited by3
Results whose statement or proof uses this declaration.
- Submodule.ClosedComplemented.isClosedproof · cited by 1
- Submodule.isTopCompl_iff_isCompl_isClosedproof · cited by 0
- ContinuousLinearMap.isStrictMap_isClosed_range_iff_quotientproof · cited by 0