A nontrivial pairing of finite T0 spaces

Abstract

There is a four-point space S 1 weakly homotopy equivalent to the circle. The restriction to S 1 of the complex number multiplication is not continuous, nevertheless a continuous model of the multiplication with values in S 1 can be defined on an eight-point circle. Applying an analogue of Hopf's construction we obtain a finite model of Hopf's famous map S 3→ S 2.