instCompleteSpaceSubtypeMemTopologicalClosure

English

The closedness property of a star-subalgebra is preserved under taking closure and star operations, yielding a closed, commutative structure when the original is commutative.

Русский

Свойство коммутативности сохраняется при замыкании и операции звезды; если исходная структура коммутативна, то замыкание тоже коммутативно.

LaTeX

$$If the original algebra is commutative on its carrier, then the topologicalClosure is a CommRing.$$

Lean4

instance {A : Type*} [UniformSpace A] [CompleteSpace A] [Semiring A] [StarRing A] [IsTopologicalSemiring A]
    [ContinuousStar A] [Algebra R A] [StarModule R A] {S : StarSubalgebra R A} : CompleteSpace S.topologicalClosure :=
  isClosed_closure.completeSpace_coe

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