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:

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>>> Let (Y,d) be a subspace of a metric space (X,d).

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>>> If (Y,d) is complete, then Y is closed with respect to d. That is,

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>>> complete==>closed.

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>>> Alternatively, if (Y,d) is complete, then Y contains all its limit

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>>> points.

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>>> Would anyone happen to know of a counterexample for the converse? That

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>>> is, does someone know of any example that demonstrates that

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>>> closed --> complete

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>>> is *not* true? I don't know for sure that it is not true, but I might

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>>> guess that it is not true.

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>>> Many thanks in advance,

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>>> Dan

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>> You need to understand that "closed" and "open" don't characterize topologies.

Actually, it is precisely the distinction of "open" *or* "closed" as

an arbitrary label on a collection of subsets satisfying the axioms

which characterizes a topology.

Using a metric to govern that specification is what makes a

topological space a metric space.

But, Paul is correct in his observations that you are conflating

terms.

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 a

characteristic of the specification of closed sets in

X.

Completion of an incomplete space is a logical type operation.

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. 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. 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. These "numbers" have no corresponding

rational 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 to

speak of these sequences as converging to a point

in the underlying set. But most authors had not

been so careful. Ultimately, it became the

essential distinction for Cantor and he used

it for the definition of a real number in

preference to Dedekind cuts.

The purpose for such care in this construction

is the identity relation. A full construction of

the reals from the natural numbers preserves the

order relation of the naturals across the type

hierarchy. Thus, the order relation of the integers

is inherited from the naturals and the order relation

of the rationals is inherited from the integers.

That there are "gaps" in the rationals follows

from the solution of polynomials that require

irrational roots. But, between any two distinct

given rationals, one can find a third rational

different from the given pair. Defining a

complete space from the rationals fills these

gaps while preserving the order relation. In

turn, trichotomy on the rationals is inherited

by the reals of the new space and the identity

relation on the reals is established.

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. The "closeness"

of a dense subset to its defining space is

expressed by the fact that it has non-empty

intersection with every open set of the

topology.

I think I got all of that right. But, there

are far more knowledgeable topologists

in this forum.