Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0904.54027
Alas, O.T.; Protasov, I.V.; Tkačenko, M.G.; Tkachuk, V.V.; Wilson, R.G.; Yaschenko, I.V.
(Tkachenko, M.G.; Yashchenko, I.V.)
Almost all submaximal groups are paracompact and $\sigma$-discrete. (English)
[J] Fundam. Math. 156, No.3, 241-260 (1998). ISSN 0016-2736; ISSN 1730-6329/e

A topological space $X=(X, {\cal T})$ is said to be maximal if every dense subset of $X$ is open, and submaximal if (a) $X$ is dense-in-itself and (b) no dense-in-itself topology on $X$ properly contains ${\cal T}$. The authors show that if $(X, {\cal T})$ is submaximal and $(X, {\cal U})$ is maximal with ${\cal U} \supseteq{\cal T}$, then: (i) $(X, {\cal T})$ and $(X, {\cal U})$ have the same dense sets, the same discrete subsets, the same cellular number, and the same Souslin number; (ii) $(X, {\cal T})$ is normal [resp., collectionwise Hausdorff, resp., paracompact] iff $(X, {\cal U})$ also is; and (iii) if ${\cal T}$ is regular then ${\cal U}$ is Tikhonov. Further, a regular submaximal space $X$ satisfies $| X|\le 2^{d (X)}$, and a submaximal $X$ with $\chi(X) =\omega_1$ satisfies $\psi(X) =\omega$.\par Turning to topological groups, the authors respond to questions posed by {\it A. V. Arkhangel'skij} and {\it P. J. Collins} [Topology Appl. 64, No. 3, 219-241 (1995; Zbl 0826.54002)] with these results: (1) If a topological group $G$ is $\kappa$-bounded in the sense that for every nonempty open $U\subseteq G$ there is $A\in [G]^{\le \kappa}$ such that $G=AU$, then $| G|\le \kappa$. (2) Every submaximal Abelian $G$ is hereditarily paracompact, strongly $\sigma$-compact, and strongly zero-dimensional; if the Abelian hypothesis is omitted, the same conclusion follows provided that the cardinal number $| G|$ is not Ulam-measurable.
[W.W.Comfort (Middletown)]

MSC 2000:: *54H11 Topological groups (topological aspects)
22A05 Structure of general topological groups
54A25 Cardinality properties of topological spaces

Citations: Zbl 0826.54002

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]