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`BookProof.ChapterSeparableSpectrum.metrizableSpace_iff_separableSpace_continuousMap` : MetrizableSpace Y ↔ SeparableSpace C(Y, ℂ)

Proved
BookProof.ChapterSeparableSpectrum.metrizableSpace_iff_separableSpace_continuousMap

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSeparableSpectrum.

BookProof.ChapterSeparableSpectrum.metrizableSpace_iff_separableSpace_continuousMap : MetrizableSpace Y ↔ SeparableSpace C(Y, ℂ)

Formalization note: Lean 4 identifier BookProof.ChapterSeparableSpectrum.metrizableSpace_iff_separableSpace_continuousMap.

Preamble
-- Generated from ChapterSeparableSpectrum.lean — theorem BookProof.ChapterSeparableSpectrum.metrizableSpace_iff_separableSpace_continuousMap
import Definitions.Def_ChapterAbelianGelfandModel
import Definitions.Def_ChapterAbelianDirectSum
import Definitions.Def_ChapterStandardBorelClassification
import Mathlib
import Definitions.Def_ChapterSeparableSpectrum
open BookProof.ChapterSeparableSpectrum


noncomputable section

open MeasureTheory TopologicalSpace WeakDual


open BookProof.ChapterAbelianGelfandModel BookProof.ChapterAbelianDirectSum
open BookProof.ChapterStandardBorelClassification

variable (Y : Type*) [TopologicalSpace Y] [CompactSpace Y] [T2Space Y]
Formal statement
theorem BookProof.ChapterSeparableSpectrum.metrizableSpace_iff_separableSpace_continuousMap :
    MetrizableSpace Y ↔ SeparableSpace C(Y, ℂ) := by sorry

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