I'm just going through Hatcher, in which he stresses there is a difference between a space being contractible & it deformation retracting to a point - but I cannot figure out what the difference is!
I realise that he gives an example of when they are different at the end of chapter 0, but I cant work out why we have a difference by just looking at the definitions.
Near the bottom of page 3, he says that a deformation retraction is "a homotopy from the identity map of X to a retraction from X onto a subspace A". In his second sentance of page 4 he says that a contractible space is one where "the identity map is homotopy to the constant map".
But surely the constant map is a retraction, and so therefore such a homotopy is always a deformation retraction. So then ANY contractible space would also deformation retract onto a point.
It seems to me that these 2 definitions are basically equivalent.
But am I wrong?
Please don't give me a counterexample - instead explain to me why these 2 definitions [contractible and deformation retracts to a point] are different.
Thank you x
