Group action on zero-dimensional spaces

Abstract

Let X be a Tychonoff space, H ( X ) the group of all self-homeomorphisms of X and e : ( f , x ) ∈ H ( X ) × X → f ( x ) ∈ X the evaluation function. Call an admissible group topology on H ( X ) any topological group topology on H ( X ) that makes the evaluation function a group action. Denote by L H ( X ) the upper-semilattice of all admissible group topologies on H ( X ) ordered by the usual inclusion. We show that if X is a product of zero-dimensional spaces each satisfying the property: any two non-empty clopen subspaces are homeomorphic, then L H ( X ) is a complete lattice. Its minimum coincides with the clopen–open topology and with the topology of uniform convergence determined by a T 2 -compactification of X to which every self-homeomorphism of X continuously extends. Besides, since the left, the right and the two-sided uniformities are non-Archimedean, the minimum is also zero-dimensional. As a corollary, if X is a zero-dimensional metrizable space of diversity one, such as for instance the rationals, the irrationals, the Baire spaces, then L H ( X ) admits as minimum the closed–open topology induced by the Stone–Čech-compactification of X which, in the case, agrees with the Freudenthal compactification of X.