```Date: Feb 2, 2013 9:41 PM
Author: Shmuel (Seymour J.) Metz
Subject: Re: looking for example of closed set that is *not* complete in a metric space

In <GbydnfG1Xq3Nu5HMnZ2dnUVZ_jqdnZ2d@giganews.com>, on 02/01/2013   at 02:32 PM, fom <fomJUNK@nyms.net> said:>So, for example, there are "gaps" in the system of rational >numbers. One can, assuming completed infinities, define infinite >sets of rational numbers corresponding to the elements of a >Cauchy sequence.What would be the point? You're introducing extra machinery into whatis a very simple construction.>When the limit of the sequence is, itself, a rational number, >that infinite set becomes a representation of that rational number >in the complete space whose "numbers" are equivalence classes of >Cauchy sequences sharing the same limit.No; the set of values taken by the sequence is irrelevant. If a Cauchysequence in the rationals converges then the *sequence* is arepresentative of its limit.>When the limit of a Cauchy sequence does not exist as a rational >number, that Cauchy sequence becomes a representative of the >equivalence class of Cauchy sequences that cannot be>differentiated from that representative using the order relation>between the rational numbers of the underlying set. What are you trying to say? The definition of the equivalence relationis the same whether the Cauchy sequences converge or not; twosequences are equivalent if their difference converges to zero. AnyCauchy sequence is a representative of its equivalence class, bydefinition.>These "numbers" have no corresponding rational number as a limit >and are, therefore, distinguished as a different logical type in >the *new*, completed space.Non sequitor, and false. There is nothing logically special aboutequivalence classes of Cauchy sequences that converge.>To call a subset of a complete space a dense subset is to say that >such a logical type construction could be made from that subset>to recover the original space.Completeness only applies to metric spaces[1], while denseness appliesto subsets of arbitrary topological spaces. You can't in generalreconstruct a topological space from only a dense subset, not even ifthe space is compact.[1] Well, slightly more general.-- Shmuel (Seymour J.) Metz, SysProg and JOAT  <http://patriot.net/~shmuel>Unsolicited bulk E-mail subject to legal action.  I reserve theright to publicly post or ridicule any abusive E-mail.  Reply todomain Patriot dot net user shmuel+news to contact me.  Do notreply to spamtrap@library.lspace.org
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