From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2769 Path: news.gmane.org!not-for-mail From: "Tom Leinster" Newsgroups: gmane.science.mathematics.categories Subject: Re: limits of finite sets Date: Sat, 31 Jul 2004 13:02:27 +0100 (BST) Message-ID: <32775.62.252.132.163.1091275347.squirrel@mail.maths.gla.ac.uk> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1241018892 5724 80.91.229.2 (29 Apr 2009 15:28:12 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:28:12 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Sat Jul 31 19:38:45 2004 -0300 X-Keywords: Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 31 Jul 2004 19:38:45 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1Br2RT-0005pA-00 for categories-list@mta.ca; Sat, 31 Jul 2004 19:35:03 -0300 X-Priority: 3 Importance: Normal X-Mailer: SquirrelMail (version 1.2.10) Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 39 Xref: news.gmane.org gmane.science.mathematics.categories:2769 Archived-At: I asked for an explanation of the following result: > the limit in Set of any diagram > > ... ---> S_3 ---> S_2 ---> S_1 > > of finite nonempty sets is nonempty Thanks very much to all who replied. I'll summarize some of the points made to me in private replies: 1. This is called Koenig's Lemma, and is usually stated in the form "any finitely-branching infinite tree contains an infinite (positively oriented) path". 2. This also follows from a general result in topology by regarding each S_n as a discrete space. The general result is that any "suitably-shaped" limit of nonempty compact Hausdorff spaces is nonempty. For Bourbaki (General Topology), "suitably-shaped" means indexed by a directed poset. More generally still, it could be any componentwise cofiltered limit, i.e. any limit for which each connected-component of the indexing category I is cofiltered (or equivalently, every finite connected diagram in I admits a cone). The proof of the general topological result specializes to give a nice topological proof of Koenig. For each n, let V_n be the subset of the product \prod_n S_n consisting of those sequences whose first n terms are compatible; then \lim_n S_n is the intersection of the (V_n)s. But with the discrete topology on each S_n, Tychonoff says that \prod_n S_n is compact, and (V_n) is a nested sequence of nonempty closed subsets so has nonempty intersection. Tom