Sets* (pointed sets). 11. If X = {xo} is a one-point space, then n1(X, x o) = {l}. Choosing a basepoint in X is only an artifice to extract a group from a groupoid. On this minor point, we have constructed new categories Top* and hTop*; eventually, we shall see that we have not overreacted. Nevertheless, these constructions raise an honest question: Do spaces having the same homotopy type have isomorphic fundamental groups?

Note that there are other possible choices for j, namely, ](t) = mt + k for any fixed integer k. ) Here is the point of these remarks. Investigation of 71:1(SI) in the spirit of the winding number suggests constructing maps 1: I -+ R with f(t) = e 2lt ;j(t) (for every closed path fin SI); moreover, attention should be paid to ](1) and ](0). 14. Let X be a compact convex subset of some R\ let f: (X, xo)-+ (Sl, 1) be continuous, let to E Z, and let exp t denote e21[it. Then there exists a unique continuous j: (X, xo) -+ (R, to) with expj = f (R, to) }/~ j "P (X, x o) ~ (Sl, 1) Remarks.