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If (C, F ) is a concrete category, then (C op , F ) is a concrete category for some (faithful) functor F : C op → Set. Proof. Let (C, F ) be a concrete category. The functor F : C → Set induces a faithful (covariant) functor F : C op → Setop (same object map and same morphism map). 3 can be regarded as a covariant functor P : Setop → Set. Claim: P is faithful. Let α, β : Y → X be morphisms in Setop and assume that α = β. Then the functions α, β : X → Y are not equal, implying that α(x) = β(x) for some x ∈ X.

4 Theorem. If (C, F ) is a concrete category, then (C op , F ) is a concrete category for some (faithful) functor F : C op → Set. Proof. Let (C, F ) be a concrete category. The functor F : C → Set induces a faithful (covariant) functor F : C op → Setop (same object map and same morphism map). 3 can be regarded as a covariant functor P : Setop → Set. Claim: P is faithful. Let α, β : Y → X be morphisms in Setop and assume that α = β. Then the functions α, β : X → Y are not equal, implying that α(x) = β(x) for some x ∈ X.

Then there exists β : F (y) → F (x) such that β F (α) = 1F (x) . Since F is full, we have β = F (β) for some β : y → x. Therefore, F (βα) = F (β)F (α) = β F (α) = 1F (x) = F (1x ) and, since F is faithful, we get βα = 1x , so that α is split monic. An analogous proof shows that F reflects split monics and hence isomorphisms. This proves (ii). 3 Example (Circle is not homeomorphic to sphere) We use the preceding theorem to show that the circle S 1 is not homeomorphic to the sphere S 2 . Suppose, to the contrary, that α : S 1 → S 2 is an isomorphism in Top.