fom
Posts:
1,037
Registered:
12/4/12
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Re: looking for example of closed set that is *not* complete in a
metric space
Posted:
Feb 1, 2013 3:32 PM
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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.
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