Why we are sure that $h^{'} $ is a homeomorphism?

Question

Here is the proof that every cofibration is an embedding from "Introduction to homotopy theory" by Martin Arkowitz :

My question is: Why we are sure that $h^{'}$ is a homomorphism? as that will imply that $f^{'}$ is a homeomorphism.

Answer 1

I think it says $i’$ is a homeomorphism instead, and this is obvious by the definition of the domain and range of $i’$ (also note that $i’$ maps the elements in the domain to the same element in the range, as $i$ was defined as the inclusion). Now with this in our mind, we can conclude that $f’$ is also a homeomorphism.