English
For any topological space X, X is simply connected if and only if X is nonempty and for all x,y in X, there exists a unique homotopy class of paths between x and y (more precisely, a unique Path.Homotopic.Quotient).
Русский
Для любой топологической_space X пространство просто связно тогда, когда X непусто и для любых точек x,y в X существует единственный класс однопроходной конъектной траектории между x и y.
LaTeX
$$$\text{SimplyConnectedSpace } X \iff (\text{Nonempty } X) \land \forall x,y \in X, \; \text{Nonempty}(\text{Unique}(Path.Homotopic.Quotient x y))$$$
Lean4
/-- An abbreviation for `prodToProdTop`, with some types already in place to help the
typechecker. In particular, the first path should be on the ulifted unit interval. -/
abbrev prodToProdTopI {a₁ a₂ : TopCat.of (ULift I)} {b₁ b₂ : X} (p₁ : fromTop a₁ ⟶ fromTop a₂)
(p₂ : fromTop b₁ ⟶ fromTop b₂) :=
(prodToProdTop (TopCat.of <| ULift I) X).map (X := (⟨a₁⟩, ⟨b₁⟩)) (Y := (⟨a₂⟩, ⟨b₂⟩)) (p₁, p₂)
