I'm trying to prove that if in (Haus, U) a morphism $f: X \rightarrow Y$ is not dense, it isn't an epimorphism. To prove this I need the following construction:
Suppose $M = \overline{f(X)}$ is closed in $Y$. We can form the coproduct $j_{1}, j_{2}: Y \rightarrow Y_{1} + Y_{2}$ of two copies of $Y$ and then identify the corresponding points in the copies of $M$ by $\varphi : Y_{1} + Y_{2} \rightarrow Z$, where $Z$ carries the final topology.
Now i can conclude that $(\varphi \circ j_{1}) \circ f = (\varphi \circ j_{2}) \circ f,$ but $\varphi \circ j_{1} \neq \varphi \circ j_{2}$.
The only problem is that to finish the proof, I need to show that the space $Z$ is a Hausdorff space.
Can anyone explain me how to do this? I think I need to work out the different options (trying to separate a point in $M$ and in $Y_{1}$, to points in $M$, etc.), but I don't know how to start my proof.