On some orthogonal factorization systems

The orthogonality relation between arrows in the class of all morphisms of a given category C yields a "concrete" antitone Galois connection between the class of all subclasses of morphisms of C. For a class Σ of morphisms of C, we denote by ^⊥ Σ(resp., Σ^⊥) the class of all morphisms f in C such that f ⊥ g (resp., g ⊥ f) for each morphism g in Σ. A couple (Σ, Γ) of classes of morphisms is said to be an (orthogonal) prefactorization system if Σ ^⊥= Γ and ^⊥ Γ = Σ. If, in addition the pfs satisfies ^⊥ Σ = Iso = Γ ^⊥, then it will be called a dense prefactorization system. A pair E,M of classes of morphisms in a category C is called an (orthogonal) factorization system if it is a prefactorization system and each morphism f in C has a factorization f = me, with e ∈ E and m ∈ M. This paper provides several examples of factorization systems and dense factorization systems in the category Top of topological spaces.