From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3921 Path: news.gmane.org!not-for-mail From: "Stephen Urban Chase" Newsgroups: gmane.science.mathematics.categories Subject: Re: Homomorphisms on Z^n Date: Tue, 18 Sep 2007 12:53:26 -0400 (EDT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019605 10898 80.91.229.2 (29 Apr 2009 15:40:05 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:05 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Sep 18 20:33:45 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 18 Sep 2007 20:33:45 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IXmQT-0005bd-II for categories-list@mta.ca; Tue, 18 Sep 2007 20:24:17 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 45 Original-Lines: 50 Xref: news.gmane.org gmane.science.mathematics.categories:3921 Archived-At: Zeeman proved the assertion in 1955 for non-measurable cardinals n. Ther= e are more general results which comprise the theory of slender (abelian) groups; see, e.g., Chapter XIII, Sections 94-95 of [L. Fuchs, Infinite Abelian Groups vol. 2, Academic Press, 1973], especially Corollary 94.6 o= n p 162. I haven't thought much about abelian groups since the early 1960's, but i= t is well known that infinite direct products have very interesting properties, both as abstract and as topological groups. For example, a closed subgroup of a direct product of countably many copies of Z is also a direct product, but not so for uncountable products (see [R.J. Nunke, On direct products of infinite cyclic groups, Proc. Amer. Math. Soc. 13 (1962), pp 66-71]). In fact, Zeeman's result implies that a countable free group is a closed subgroup of a direct product with uncountably many factors. A generalization of Nunke's theorem and some related results are contained in my old paper [Function topologies on abelian groups, Ill. J. Math. 7 (1963), pp 593-608]. Steve Chase ---------------------------- Original Message ---------------------------= - Subject: categories: Homomorphisms on Z^n From: "Michael Barr" Date: Fri, September 14, 2007 8:34 am To: "Categories list" -------------------------------------------------------------------------= - Many years ago (at least 45) Harrison mentioned to me that for any n (including infinite cardinals), Hom(Z^n,Z) =3D n.Z, in other words the Z-dual of the product is the sum. This is obviously a very special property of Z, almost the negation of injectivity. Has anyone on this list ever seen this before and can give me a reference? Michael