fom
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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


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" 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 nonempty 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.

