Group action on zero-dimensional spaces

摘要

Let X be a Tychonoff space, H(X) the group of all self-homeomorphisms of X and e: (d, x) ∈ H(X) 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-Cech-compactification of X which, in the case, agrees with the Freudenthal compactification of X.