Date: Sun, 17 Jan 1999 14:11:01 -0500 (EST) From: Michael Cole Subject: Re: question abt basepts Claim: If X and Y are based homotopy equivalent and X is nondegenerately based then Y is not necessarily nondegenerately based. To make the counterexample, let X=*. We need to find a based contractible space Y that has a degenerate basepoint. I think the following works. Let Y' be the subspace of the reals consisting of 0 and 1/n for n \geq 1 with the basepoint at 0. Let Y be the reduced cone of Y'. Then Y is based contractible, but the basepoint is degenerate. Mike Cole _________________________________________________________________ From: Gerd Laures Date: Mon, 18 Jan 99 11:04:26 +0100 Subject: Re: question abt basepts Tony Elmendorf wrote: Let X and Y be based spaces which are based homotopy equivalent, and suppose X has a nondegenerate basepoint. Prove or give a counterexample: Y must also have a nondegenerate basepoint. Answer: Here is a counterexample: Let M be a non countable set and Y be the product of intervals I^M with basepoint pt={o}^M. Y is based homotopy equivalent to X=pt. X is well pointed but Y is not. This example is taken from tomDiek, Kamps, Puppe: Homotopietheorie, LNM 157, p. 78/79. Cofibrations are not invariant under homotopy equivalences ``under A'' but h-cofibrations are. Gerd Laures