  We show that the Riemann zeta function \zeta\ has only countably many
self-intersections on the critical line, i.e., for all but countably many z in
C the equation \zeta(1/2+it)=z has at most one solution t in R. More generally,
we prove that if F is analytic in a complex neighborhood of R and locally
injective on R, then either the set {(a,b) in R^2:a \ne b and F(a)=F(b)} is
countable, or the image F(R) is a loop in C.
