```Date: Feb 1, 2013 3:32 PM
Author: fom
Subject: Re: looking for example of closed set that is *not* complete in a<br> metric space

On 2/1/2013 12:09 PM, pepstein5@gmail.com wrote:> On Friday, February 1, 2013 4:52:55 PM UTC, peps...@gmail.com wrote:>> On Friday, February 1, 2013 4:37:40 PM UTC, Daniel J. Greenhoe wrote:>>>>> Let (Y,d) be a subspace of a metric space (X,d).>>>>>>>>>>>>>>>>>>>> If (Y,d) is complete, then Y is closed with respect to d. That is,>>>>>>>>>>>>>>>>>>>>    complete==>closed.>>>>>>>>>>>>>>>>>>>> Alternatively, if (Y,d) is complete, then Y contains all its limit>>>>>>>>>> points.>>>>>>>>>>>>>>>>>>>> Would anyone happen to know of a counterexample for the converse? That>>>>>>>>>> is, does someone know of any example that demonstrates that>>>>>>>>>>     closed --> complete>>>>>>>>>> is *not* true? I don't know for sure that it is not true, but I might>>>>>>>>>> guess that it is not true.>>>>>>>>>>>>>>>>>>>> Many thanks in advance,>>>>>>>>>> Dan>>>>>>>> You need to understand that "closed" and "open" don't characterize topologies.Actually, it is precisely the distinction of "open" *or* "closed" asan arbitrary label on a collection of subsets satisfying the axiomswhich characterizes a topology.Using a metric to govern that specification is what makes atopological space a metric space.But, Paul is correct in his observations that you are conflatingterms.Y would always be closed as topological space in its own right.That is a property of the defining axioms.Whether or not Y is closed in X as a subset of X is acharacteristic of the specification of closed sets inX.Completion of an incomplete space is a logical type operation.So, for example, there are "gaps" in the system of rationalnumbers.  One can, assuming completed infinities, defineinfinite sets of rational numbers corresponding to theelements of a Cauchy sequence.  When the limit of thesequence is, itself, a rational number, that infiniteset becomes a representation of that rational number inthe complete space whose "numbers" are equivalence classesof Cauchy sequences sharing the same limit.  When thelimit of a Cauchy sequence does not exist as a rationalnumber, that Cauchy sequence becomes a representativeof the equivalence class of Cauchy sequences that cannotbe differentiated from that representative using theorder relation between the rational numbers of theunderlying set.  These "numbers" have no correspondingrational number as a limit and are, therefore,distinguished as a different logical type in the *new*,completed space.Apparently, Cauchy had been very careful not tospeak of these sequences as converging to a pointin the underlying set.  But most authors had notbeen so careful.  Ultimately, it became theessential distinction for Cantor and he usedit for the definition of a real number inpreference to Dedekind cuts.The purpose for such care in this constructionis the identity relation.  A full construction ofthe reals from the natural numbers preserves theorder relation of the naturals across the typehierarchy.  Thus, the order relation of the integersis inherited from the naturals and the order relationof the rationals is inherited from the integers.That there are "gaps" in the rationals followsfrom the solution of polynomials that requireirrational roots.  But, between any two distinctgiven rationals, one can find a third rationaldifferent from the given pair.  Defining acomplete space from the rationals fills thesegaps while preserving the order relation.  Inturn, trichotomy on the rationals is inheritedby the reals of the new space and the identityrelation on the reals is established.To call a subset of a complete space a densesubset is to say that such a logical typeconstruction could be made from that subsetto recover the original space.  The "closeness"of a dense subset to its defining space isexpressed by the fact that it has non-emptyintersection with every open set of thetopology.I think I got all of that right.  But, thereare far more knowledgeable topologistsin this forum.
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