A Class of Compact Rigid 0-Dimensional Spaces

Extract

A topological space is called “rigid” if its autohomeomorphism group is trivial. In (1), de Groot and McDowell showed that there are rigid, 0- dimensional spaces of arbitrarily high cardinality but left open the question of whether or not there are compact,rigid, 0-dimensional spaces of arbitrarily high cardinality, pointing out that an affirmative answer implies the existence of arbitrarily large Boolean rings with trivial automorphism groups. In this paper we construct a class of rigid, 0-dimensional spaces X ^αof arbitrary infinite cardinality and show that their Stone-Cech compactifications βX ^αare also rigid, thus answering the above question affirmatively.

I would like to thank S. W. Willard, J. R. Isbell, and the referee for their careful readings of preliminary versions of this paper.