LET US CONSIDER K + 1 COPIES OF THE UNION AS SHOWN ON THE PICTURE.

A. T. Fomenko, D. B. Fuchs, and V. L. Gutemacher, Homotopic Topology, Akademiaia Kiado, Budapest, 1986

At first we take the first letter of the word. It corresponds to some circle in the union. We cut out a small section of this circle in the first and second copies. Then we unite the free ends crosswise. the projection is defined on the modified space in the obvious way. Next we connect the second and third copies similarly, this time by using the second letter in the word. We go on with this procedure until we get a connected space with a projection on the union.