Continuous functions on Hausdorff continua

摘要

We consider the question: given a set X and a function T : X -> X, does there exist a topology on X with respect to which X is a Hausdorff continuum and T is continuous? We answer the question for the class of kappa-to-one surjective functions (kappa >= c) with finitely based orbit spectrums. We show that if a member T : X -> X of the class is compactifiable (X can be equipped with a compact Hausdorff topology with respect to which T continuous), then there is a Hausdorff continuum Y and a continuous function T' : Y -> Y with the same orbit spectrum as that of T. (C) 2016 Elsevier B.V. All rights reserved.