A Map of a Polyhedron onto a Disk

A map f: X → Y is said to be universal if for every map g:X → Y there exists an x ∈ X such that f(x) = g(x). In [2] W. Holsztynski observed that if B is a Boltyanskiĭ continuum (see [1]), then there exists a universal map f:B→ I 2 such that the product map fxf:BxB→I 2×I 2 is not universal. Using this he showed that B can be replaced by a two-dimensional polyhedron. He did not, however, give a concrete example. We exhibit explicitly a two-dimensional polyhedron K and a universal map f:K→ I 2 such that f×f:K×K→ I 2×I 2 is not universal.