```Date: Feb 3, 2013 6:40 AM
Author: fom
Subject: Re: looking for example of closed set that is *not* complete in a<br> metric space

On 2/2/2013 8:41 PM, Shmuel (Seymour J.) Metz wrote:> 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 what> is a very simple construction.That is not me.  I would be happy to understandthe real numbers in the simple manner inwhich it had been taught to me.The construction of the reals from the natural numbersis a sequence of logical types for which the order relationof the natural numbers grounds the order relation of thederived type.You can find an excellent construction of the integers inJacobson:Lectures in Abstract AlgebraVan Nostrand Co., Inc.Princeton NJ, c. 1951Each integer is an infinite class of pairs. From this, the similar construction for definingthe rationals should be apparent using quotientsrather than differences.  Some particularmay be needed for handling 0.Once again, each rational is an infinite class of pairs. From this, one constructs the reals.  The followingis from Cantor's Grundlagen concerning the logicof definition for a real number:=============================="I come now to the third definitionof real numbers.  Here too an *infinite*set of rational numbers of the firstpower is taken as a basis, but it hasproperties other than the Weierstrassiandefinition. [...]"Every such set (a_v), which also canbe characterized by the requirement:lim_(v=oo) (a_(v+u) - a_v) = 0 (for arbitrary u)I call a fundamental sequence and correlatewith it a number b, *TO BE DEFINED THROUGHIT*, [...]"Care must be taken on this cardinal point,whose significance can easily be overlooked:in the third definition the number b, say,is *NOT DEFINED AS THE 'LIMIT' OF THE TERMSa_v OF A FUNDAMENTAL SEQUENCE (a_v); FORTHIS WOULD BE A LOGICAL ERROR SIMILAR TOTHE ONE DISCUSSED FOR THE FIRST DEFINITION,i.e., WE WOULD BE PRESUMING THE EXISTENCEOF THE LIMIT lim_(v=oo) a_v.  BUT THESITUATION IS RATHER THE REVERSE.and earlier..."These definitions agree that anirrational real number is *GIVEN BYA WELL-DEFINED INFINITE SET OF RATIONALNUMBERS* of the first power.  But theydiffer over the way in which the setis linked with the number it defines,and in the conditions which the sethas to fulfill in order to qualify asa foundation for the definition inquestion."In the first definition a set ofpositive rational numbers a_v istaken as a basis, is designated by(a_v), and satisfies the conditionthat, whatever and however many ofthe a_v are summed (so long as thecount is finite) this sum alwaysremains less than a specifiablelimit.  [...]"One sees here that the creativeelement which binds the set withthe number defined through it liesin the formation of sums; but it mustbe emphasized that only the summationof an always finite count of rationalelements is used *AND THAT THE NUMBERb TO BE DEFINED IS NOT SET AT THEBEGINNING AS EQUAL TO THE SUM Sum_v(a_v)OF THE INFINITE SERIES (a_v); THISWOULD BE A LOGICAL ERROR,[...]=================By this account, the real numbers are alogical type wherein each individual realnumber is an equivalence class of fundamentalsequences.I have no doubt that I am "wrong" in theeyes of many.  But it would be nice tohave that error explained.At what point did we start ignoring the "foundations"of mathematics because the logical rigor wasexcessive?  Believe me.  I get the usual axiomatizationof the real numbers as presented in any decent calculustext.  But strict logical form that places thefoundations of mathematics within set theory usingonly the axioms of set theory obtains each real numberthrough a sequence of definitions describing classesof distinct logical types.I am fully aware that much research on thereal numbers in set theory involves the isomorphicset omega^(omega) which is closely relatedto the description of real numbers relative toinfinite continued fractions (Baire space).But, the sets involved in this isomorphism donot have members identifiable as real numbersrelative to construction using continuedfractions.  Such a definition would alsoinvolve equivalence classes of infinitecollections of rationals.Now, I certainly did make the mistakeof thinking that the original posterhad less knowledge than subsequent postsindicated.  But, to the best of myknowledge, my remarks have been accurateand to the point with regard to whatmodern mathematicians claim to believewhen they say the things they say concerningsets -- especially ZFC.>>> 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 Cauchy> sequence in the rationals converges then the *sequence* is a> representative of its limit.>You are making precisely the logical error spoken of by Cantor.>> 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 relation> is the same whether the Cauchy sequences converge or not; two> sequences are equivalent if their difference converges to zero. Any> Cauchy sequence is a representative of its equivalence class, by> definition.This is the point at which the order relation of the underlyingset of rationals is used to ground the identity relation for theequivalence classes of objects defined individually as infinite collections of rationals.>>> 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 about> equivalence classes of Cauchy sequences that converge.Sadly I only speak English.  But the false part I takeissue with.  The entire problem since the discovery ofincommensurables has been to find a satisfactory logicalexplanation for them.  In Dedekind's definition using cuts,there certainly is a logical difference between cuts containinga maximum value (or minimum depending on which is used) andthose that do not.And Cantor's careful distinctions have been made preciselyto accommodate the *DEFINITION* of real numbers, both rationaland irrational.>>> 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 applies> to subsets of arbitrary topological spaces. You can't in general> reconstruct a topological space from only a dense subset, not even if> the space is compact.>> [1] Well, slightly more general.>As I said when I concluded before, there are certainlymore knowledgeable topologists than I.Yet,"A uniform space is complete if and onlyif every Cauchy net in the space convergesto a point of the space""A uniform space is complete if and onlyif each family of closed sets which hasthe finite intersection property andcontains small sets has a non-voidintersection""A pseudo-metrizable uniform space iscomplete if and only if every Cauchysequence in the space converges to apoint"And,"A set A is dense in a topologicalspace X iff the closure of A is X"All of these statements are from GeneralTopology by Kelley.  The first and fourthare definitions.  Both contradict whatyou have just stated.  What you are thinkingof as a dense subset is a set dense in asubset with the subspace topology.As I said, I am probably wrong.  But I tryto be careful.
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