Hi,

In my course topology there's a propisition which says the following: suppose $\displaystyle f$ and $\displaystyle g$ are continuous functions and $\displaystyle f \circ g$ is initial then $\displaystyle g$ is initial (I mean with initial that $\displaystyle f \circ g$ and $\displaystyle g$ have the initial topology on their domains).

Now, they ask for a counterexample. I have to give an example where $\displaystyle g$ and $\displaystyle f \circ g$ are initial but $\displaystyle g$ is not a continuous function.

The problem is that I don't have an idea how to construct an initial function.

Anyone?