Topologically generating weight of compact groups
by
Dmitri Shakhmatov
Department of Mathematics, Faculty of Science, Ehime University, Matsuyama 790, Japan
Coauthors: Dikran Dikranjan (Department of Mathematics and Computer Science, Udine University, Udine, Italy)

We use w(X) to denote the weight of X, i.e. the smallest size of a base for the topology of X if such a base is infinite, or \omega otherwise.

Let G be a compact Hausdorff group. A subspace X of G topologically generates G if the smallest subgroup of G that contains X is dense in G. Define tgw(G) to be the minimim cardinaility of the weight w(F) of a closed subset F of G topologically generating G. Clearly tgw(G) <= w(G).

For a compact group G we completely compute the value tgw(G) of ``topologically generating weight '' of G:

Theorem 1. (i) tgw(G) = w(G) in case G is totally disconnected,

(ii) tgw(G) = omegaroot(w(G)) in case G is connected, where omegaroot(\sigma) is defined as the smallest infinite cardinal \tau such that \sigma <= \tau^\omega,

(iii) tgwG)=w(G/c(G)) × omegaroot(w(c(G))) for a (general) compact Hausdorff group G, where c(G) is the connected component of G.

While tgw(G) need not coincide with w(G), the following dichotomy holds for a connected compact Hausdorff group G of regular weight w(G): either tgw(G)=w(G) or cf(tgw(G))=\omega. We demonstrate that both cases happen ``cofinally often'': for every cardinal \sigma there exist succsessor (thus regular) cardinals \tau > \sigma and \kappa > \sigma such that cf(tgw(T^\tau))=\omega and tgw(T^\kappa)=\kappa. (Here T is the torus group.)

A super-sequence is an infinite compact space with a single non-isolated point. (Countable super-sequences are precisely (the images of) the usual convergent sequences.) Hofmann and Morris have proved that every compact Hausdorff group G can be topologically generated by a suitable super-sequence, thereby allowing one to define the cardinal s(G) as the minimum cardinality |S| of a super-sequence S topologically generating G. Clearly one has tgw(G) <= s(G) <= w(G). We prove that, rather unexpectedly, the first inequality is indeed an equality:

Theorem 2. tgw(G)=s(G) for all compact Hausdorff groups G.

Date received: June 30, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeh-54.