I'm reading "Memento on cell complexes" and the following is supposed to be a counter example of a cell complex because $e^1$ is not homeomorphic to the open segment:
To me this looks like a deformed disk and is therefore homeomorphic to $D^2$. Why is $e^1$ not homeomorphic to the open segment? Or is it missing a second black dot to denote touching of $e^1$ at 12 o'clock?
Also, I don't see why the following isn't a cell complex either. The boundary of $e^2$ maps into $X^1$. The inductive construction of a cell complex requires exactly that.
Thanks for your help!