There are arbitrarily large uniquely homogeneous spaces

摘要

A topological space X is uniquely homogeneous if for every a, b is an element of X, there is exactly one homeomorphism h : X - X such that h(a) = b. We say that X is strongly uniquely homogeneous if it is uniquely homogeneous and the only continuous functions f : X - X are homeomorphisms and constant functions. We show that if n = 2, then there is a strongly uniquely homogeneous subspace of R-n, all of whose homeomorphisms are restrictions of isometries of R-n (rigid motions if n is even). Using a similar construction, we show that if kappa is an infinite cardinal, then there is a strongly uniquely homogeneous Hausdorff and completely regular space of cardinality 2(kappa). (C) 2020 Published by Elsevier B.V.