Generalized Alexandroff–Urysohn squares and a characterization of the fixed point property

Given a Hausdorff continuum X, we introduce a topology on X × X that yields a Hausdorff continuum. We call the resulting space the Alexandroff–Urysohn square of X and prove that X has the fixed point property if and only if the Alexandroff–Urysohn square of X has the fixed point property.