I came across this assertion:
There is an epimorphism $X \overset{f}\to Y\;$ in Top such that the homotopy class $X \overset{\tilde{f}}\to Y\;$ of $f$ is not an epimorphism in hTop.
Then, by way of elaboration, the text briefly refers to "the covering projection of the real line onto the circle, defined by: $x \mapsto e^{ix}$."
I interpret this to mean that $f\;$ is $x \mapsto e^{ix}$, so $X = \mathbb{R}$, and $Y = S^1$. If this interpretation is correct, I can see that this $f:\mathbb{R}\to S^1$ is an epimorphism in Top, but I still don't see how to complete the alluded-to counterexample.
More specifically, if I understand the situation correctly, the claim above amounts to asserting that there exist a topological space $(Z, \tau\,)$ and a pair of Top morphisms $g_1, g_2:S^1 \to Z$ such that all the following hold:
- $\tilde{g_1}\;\;\tilde{\scriptstyle\circ}\;\;\tilde{f}\,\;=\;\;\tilde{g_2} \;\;\tilde{\scriptstyle\circ}\;\;\tilde{f}\;\;$
- $\tilde{g_1}\,\;\neq\;\;\tilde{g_2}\;\;$
- $g_1\;\;{\scriptstyle\circ}\;\;f\,\;\neq\;\;g_2\;\;{\scriptstyle\circ}\;\;f\;\;\;$ (otherwise, $f\;$ being epic would imply $g_1 = g_2$, and this would lead to a contradiction with (2))
The fact that $f\;$ is surjective makes it very difficult for me to envision a situation in which both (1) and (2) could possibly hold. (Also, if truth be told, maybe I have no business fighting with this problem, given that my familiarity with hTop doesn't go much beyond having read the definitions needed to follow the above.)
Any suggestions for suitable $(Z, \tau\,)$, $g_1$, and $g_2$ would be much appreciated.