locallyConstantIsoContinuousMap — Mathlib

English

Every q belongs to the trace set trStmts₁(q).

Русский

Каждое q принадлежит множеству трасс trStmts₁(q).

LaTeX

$$$\\forall q,\\; q \\in \\operatorname{trStmts}_1(q)$$$

Lean4

/-- Locally constant maps are the same as continuous maps when the target is equipped with the discrete
topology
-/
@[simps]
def locallyConstantIsoContinuousMap (Y X : Type*) [TopologicalSpace Y] :
    LocallyConstant Y X ≅ C(Y, TopCat.discrete.obj X) :=
  letI : TopologicalSpace X := ⊥
  haveI : DiscreteTopology X := ⟨rfl⟩
  { hom := fun f ↦ (f : C(Y, X))
    inv := fun f ↦ ⟨f, (IsLocallyConstant.iff_continuous f).mpr f.2⟩ }

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