Topology IV.b - homotopy theory

Homework/Semi-flipped teaching. I would like you to read the beginning of the section 6 about pushouts, alternatively watch the second half of the video from Week 9:

https://mediaspace.epfl.ch/media/Whitehead+Theorem%2C+Lecture09.2/0_gzcv14yl

It starts after 26 minutes and ends at minute 44. The reason for that is that I have to finish the proof of the uniqueness of the CW-approximation. In fact I will explain in more details the construction of the comparison map between two CW-approximations and then prove it's a weak equivalence. Then we will see that it is even a homotopy equivalence.

After that I will probably diverge somewhat from the notes because I think that left properness is not well explained and I'm not sure I can explain it well without spending much more time and technicalities.