English
There is a homeomorphism between X and the character space of C(X, 𝕜) given by the evaluation map x ↦ ev_x, in the compact Hausdorff setting.
Русский
Существует гомоморфизм-гиоморфизм между X и характерным пространством C(X, 𝕜), заданный отображением оценки x ↦ ev_x, в случае компактности и Хаусдорфа.
LaTeX
$$$\\text{homeoEval}(X, 𝕜) : X \\simeq_t (\\mathrm{WeakDual}.\\mathrm{CharacterSpace}\\; 𝕜\\; C(X, 𝕜)).Elem$$$
Lean4
/-- This is the natural homeomorphism between a compact Hausdorff space `X` and the
`WeakDual.characterSpace 𝕜 C(X, 𝕜)`. -/
noncomputable def homeoEval : X ≃ₜ characterSpace 𝕜 C(X, 𝕜) :=
@Continuous.homeoOfEquivCompactToT2 _ _ _ _ _ _
{ Equiv.ofBijective _ (continuousMapEval_bijective X 𝕜) with toFun := continuousMapEval X 𝕜 }
(map_continuous (continuousMapEval X 𝕜))
