Weight of closed subsets topologically generating a compact group

✍ Scribed by Dikran Dikranjan; Dmitri Shakhmatov

• Publisher: John Wiley and Sons
• Year: 2007
• Tongue: English
• Weight: 258 KB
• Volume: 280
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200410499

✦ Synopsis

Abstract

Let G be a compact Hausdorff group. A subspace X of G topologically generates G if G is the smallest closed subgroup of G containing X. Define

tgw (G) = ω · min{w (X): X is closed in G and topologically generates G },

where w (X) is the weight of X, i.e., the smallest size of a base for the topology of X. We prove that:

(i) tgw (G) = w (G) if G is totally disconnected,

(ii) tgw (G) = $ \root \omega \of {w(G)} $ = min{τ ≥ ω: w (G) ≤ τω } in case G is connected, and

(iii) tgw (G) = w (G /c (G)) · $ \root \omega \of {w(c(G))} $, where c (G) is the connected component of G.

If G is connected, then either tgw (G) = w (G), or cf(tgw (G)) = ω (and, moreover, w (G) = tgw (G)^+^ under the Singular Cardinal Hypothesis).

We also prove that

tgw (G) = ω · min{|X |: X ⊆ G is a compact Hausdorff space with at most one non‐isolated point topologically generating G }. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)