An exercise in Hilton/Stammbach's book begins: "Show that $j : X_0 \to X$ in $\mathfrak{T}$ [the category of topological spaces and continuous maps] is a homeomorphism of $X_0$ into $X$ if and only if it is a monomorphism". I have a couple questions about this:
(1) By "homeomorphism into" do they mean "topological embedding"? This is an old book so I wouldn't be surprised if terminology has changed since the time it was written.
(2) I already showed in an earlier exercise that the monomorphisms in $\mathfrak{T}$ are the injective continuous maps, not the topological embeddings, contrary to what's being asked of me here. Am I misreading the question?
I figured it out, I was just misreading the sentence structure. The full sentence is "Show that $j : X_0 \to X$ is a homeomorphism of $X_0$ into $X$ [topological embedding] if and only if it is a monomorphism and for any $f : Y \to X$ in $\mathfrak{T}$, a factorization $U(j)g_0 = U(f)$ in $\mathfrak{S}$ [the category of sets] implies $jf_0 = f$ in $\mathfrak{T}$ with $g_0 = U(f_0)$." It's meant to be read "Show (P iff (Q and R))", but I mistakenly read it as "(Show (P iff Q)) and (Show R)".