Description:
<jats:title>Abstract</jats:title><jats:p>A theorem by H. Robbins shows that every closed and non-empty subset of the unit ball B<jats:sup>n</jats:sup> in Euclidean n-space is the fixed point set of a self map of B<jats:sup>n</jats:sup>. This result is extended to coincidence producing maps of Bn, where a map <jats:italic>ƒ</jats:italic>:X → Y is coincidence producing (or universal) if it has a coincidence with every map g:X → Y. The main result implies that if ƒ:B<jats:sup>n</jats:sup>, S<jats:sup>n - 1</jats:sup> → B<jats:sup>n</jats:sup>, S<jats:sup>n - 1</jats:sup> is coincidence producing and A⊂B<jats:sup>n</jats:sup> closed and nonempty, then there exist a map ƒ': B<jats:sup>n</jats:sup>, S<jats:sup>n - 1</jats:sup> → B<jats:sup>n</jats:sup>, S<jats:sup>n - 1</jats:sup> and a map g: B<jats:sup>n</jats:sup> → B<jats:sup>n</jats:sup> such that ƒ' | S<jats:sup>n - 1</jats:sup> is homotopic to ƒ | S<jats:sup>n-1</jats:sup> and A is the coincidence set of <jats:italic>ƒ</jats:italic>' and g.</jats:p>