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 what

is 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 Cauchy

sequence in the rationals converges then the *sequence* is a

representative 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 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.

>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.

>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.

--

Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>

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