TopCat.mono_iff_injective
theorem TopCat.mono_iff_injective : ∀ {X Y : TopCat} (f : X ⟶ Y), CategoryTheory.Mono f ↔ Function.Injective ⇑f := by
sorry
This theorem states that for any two objects `X` and `Y` in the category of topological spaces (`TopCat`), a morphism `f` (which in this context is a continuous function) from `X` to `Y` is a monomorphism in the category theory sense if and only if it is injective as a function. In other words, a continuous map between topological spaces is a category-theoretic monomorphism precisely when it has the property that equal outputs imply equal inputs.
More concisely: A continuous function between topological spaces is a monomorphism in the category of topological spaces if and only if it is an injective function.
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TopCat.epi_iff_surjective
theorem TopCat.epi_iff_surjective : ∀ {X Y : TopCat} (f : X ⟶ Y), CategoryTheory.Epi f ↔ Function.Surjective ⇑f := by
sorry
This theorem establishes a correspondence between two concepts in different areas of mathematics: category theory and set theory. For any two objects `X` and `Y` in the category of topological spaces `TopCat` and any morphism `f` from `X` to `Y`, the theorem states that `f` is an epimorphism in the categorical sense if and only if the underlying function of `f` is surjective. In other words, a continuous map between topological spaces is categorically "onto" (epimorphic) if and only if every point in the target space is the image of some point in the source space under the map (surjective).
More concisely: In the category of topological spaces, a continuous map is an epimorphism if and only if its underlying function is surjective.
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