We consider the following problem: given a set X and a function T : X → X, does there exist a compact Hausdorff topology on X which makes T continuous? We characterize such functions in terms of their orbit structure. Given the generality of the problem, the characterization turns out to be surprisingly simple and elegant. Amongst other results, we also characterize homeomorphisms on compact metric spaces. © 2005 Elsevier Inc. All rights reserved.
Good, C., Greenwood, S., Knight, R. W., McIntyre, D. W., & Watson, S. (2006). Characterizing continuous functions on compact spaces. Advances in Mathematics, 206(2), 695–728. https://doi.org/10.1016/j.aim.2005.11.002