# Characterisation of extreme mono- and epi- morphisms in category of topological spaces.

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.

(1) It suffices to prove that $\chi$ is an open map. Suppose that $V$ is an open set in $X$ such that $A=\chi[V]$ is not open in $Z$. Let $B=\psi[A]\subseteq\varphi[X]$. Clearly $V=(\psi\circ\chi)^{-1}[B]=\varphi^{-1}[B]=(\varphi^0)^{-1}[B]$, and $\varphi^{0}$ is a homeomorphism, so $B=\varphi^0[V]$ must be open in $\varphi[X]$. It follows that $\psi^{-1}[B]$ is open in $Z$. If $\psi^{-1}[B]=A$, we’re done. If not, there must be some $z\in\psi^{-1}[B]\setminus A$. Let $x$ be the unique element of $X$ such that $\chi(x)=z$. Then $x\notin V$, but $\varphi^0(x)=(\psi\circ\chi)(x)=\psi(z)\in B=\varphi^0[V]$, contradicting the assumption that $\varphi^0$ is bijective.
(2) Assuming that you actually want to prove that $\chi:Z\to Y$ is a homeomorphism, you can use the same idea. Again it suffices to show that $\chi$ is open, so let $[V]$ be an open subset of $Z$, let $A=\chi[V]$, and let $W=\psi^{-1}[V]$, an open set in $X$. Clearly $A=\chi[V]=(\chi\circ\psi)[W]=\varphi[W]$. Let $\tau$ be the topology on $Y$. If $A\notin\tau$, the topology on $Y$ generated by $\tau\cup\{A\}$ would be a strictly stronger topology with respect to which $\varphi$ was continuous.