I am currently really stuck and confused about the following problem from Topology II of Encyclopaedia of Mathematical Sciences (Springer, Novikov/Fuchs):

Give an example of two path-connected spaces that are weak homotopy equivalent but not homotopy equivalent.

The problem is that I am not sure if there are such examples at all considering the Serre fibration lacking for path-connected spaces or their basis. Could you enlighten me?

  • $\begingroup$ Hint: can you come up with an example of a weakly-contractible space (i.e. all homotopy groups vanish) which is not contractible? By Whitehead's Theorem such an example cannot be a CW complex, so think of the weird/pathological spaces you know. $\endgroup$
    – William
    May 4, 2020 at 14:43
  • $\begingroup$ @William unfortunately I could not find any. What are those? $\endgroup$
    – user783574
    May 4, 2020 at 22:31