I have two seemingly simple questions in topology but I have no idea how to answer them.
1) Let $\varphi:X\to Y$ be a continuous map such that $\varphi^0:X\to\mathrm{Im}(\varphi)$ is a homemorphism. Assume we have a representation $\varphi=\psi\circ\chi$, where $\chi:X\to Z$ is a bijective continuous map, and $\psi:Z\to Y$ is a continuous map. How to prove that $\chi:X\to Z$ is a homeomorphism? Maps for which such an implication holds are called extreme monomorphisms.
2) Let $\varphi:X\to Y$ be a continuous map and topology of $Y$ is the strongest topology for which $\varphi$ is continuous. Assume we have a representation $\varphi=\chi\circ\psi$, where $\chi:Z\to Y$ is a bijective continuous map, and $\psi:X\to Z$ is a continuous map. How to prove that $\chi:X\to Z$ is a homeomorphism? Maps for which such an implication holds are called extreme epimorphisms.
Thanks for your help.