A Boolean algebra without proper automorphisms
Bjàrni Jónsson · Proceedings of the American Mathematical Society · 1951
BJARNIJÓNSSONIt is the purpose of this note to show that there exists an infinite Boolean algebra which has no proper automorphisms.1We shall construct a simply ordered set S, introduce a topology on this set in the usual manner, the so-called interval topology determined by the ordering relation,2 and prove that 5 is a compact zero-dimensional Hausdorff space and that the only homeomorphism on 5 onto 5 is the identity mapping.It is well known3 that the group of automorphisms of the set-field 33 consisting of all open and closed subsets of 5 is isomorphic to the group of all homeomorphisms on 5 onto S, whence it follows that 33 has no proper automorphisms.Consider a simply ordered set 5 with at least two elements.By the interval topology on 5 we mean the topology which has as a subbasis for open sets, the family of all sets UQS such that either U = {x | x E S and x < a] or U = {x\ x ES and a < -x} for some aES.These sets together with all sets of the form U = {x | x G S and a < x < &} with a, bES constitute a basis for the interval topology on 5. We shall need the following theorem.