(a) the surface of a cylinder offin

(a) the surface of a cylinder offinite height, excluding the two circles at the ends; (b) the one-sheeted hyperboloid given by the equation x2 + y2 -Z2 = 1; (c) the open annulus in the complex plane specified by 1 < I z I < 3; (d) the sphere with the points at the north and south poles removed. We propose to give a specific homeomorphism (i.e., a continuous, one-one, and onto function which has continuous inverse) from space (b) to space (c). It is most convenient to specify the points of(b) by cylindrical polar coordinates (r, e, z) and to use plane polar coordinates (r, e) for space (c). When e = 0 in (b) we obtain a branch of the hyperbola x2 -Z2 = 1, and we plan to send this nicely onto the corresponding piece ofthe annulus, i.e., the ray {(x,y) 11 < x< 3, y =: O}. If we can do a similar trick for each value of e, in a continuous manner as evaries from 0 to 2n, we shall have the required homeomorphism. Define f:(-oo, (0)~(1,3) by f(x) = x/(1 + lxi) + 2; thenfis a bijection, is con-