Question about the composite of a homeomorphism and a continuous onto function.

Question

If $f : (G,T)$ homeomorphically to $(A,T_1)$, and $h: (A,T_1)$ continuously and onto $(C,T_2)$, then is it always the case that, given the composition $g = h \circ f : (G,T) \rightarrow (C,T_3)$, the continuous onto image of $G = C$? Or $g(G) = C$?

I'm a little confused because I thought that the composite of two bijective functions did this?

Answer 1

Yes. Homeomorphisms are surjective by definition, so $$g(G)=h(f(G))=h(A)=C.$$ Also, a composition of continuous functions is always continuous.