On rigidity and groups of homeomorphisms

Abstract

If X is an object of a category, we denote by A(X) the group of all automorphisms of X. Thus for example, if X is a topological space, A(X) is the group of all homeomorphisms on X. In this paper, we investigate A(X) for objects in the categories of topological spaces and apply the results to obtain corresponding theorems in the categories of lattices, Boolean algebras, semigroups, graphs, and some other related categories. The groups of homeomorphisms have been investigated by J. de Groot [7] (see also [1], [5], [8] and [9]), the groups of automorphisms of lattices by G. Birkhoff [4] (see also [15]) and the groups of automorphisms of Boolean algebras by M. KatStov [13] and J. de Groot [7] (see also [12], [17]). Here we solve some open problems on this topic, generalize many known results and give different proofs for several results. The following are some of the main theorems of this paper: