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Topic: looking for example of closed set that is *not* complete in a metric space
Replies: 26   Last Post: Feb 3, 2013 11:06 AM

 Messages: [ Previous | Next ]
 Achimota Posts: 255 Registered: 4/30/07
Re: looking for example of closed set that is *not* complete in a
metric space

Posted: Feb 2, 2013 12:10 AM

On Saturday, February 2, 2013 12:52:55 AM UTC+8, peps...@gmail.com wrote:
> ...To say that a space is "closed"
> (as in your statement "closed -> complete") doesn't really mean anything.
> To make progress replace "closed -> complete" by something more
> formal and rigorous and precise.

This is certainly good advice and many apologies for my sloppy original posting. Is the following any better?...

Let (X,d) be a metric space.
Let T be the topology induced by d and
(X,T) be the resulting topological space.
Let Y be a subset of X.
Then
(Y,d) is complete ==> Y is closed in (X,d).
Alternatively,
(Y,d) is complete ==> Y is closed in (X,T).

But what about the converse? That is, is this true?
Y is closed in (X,d) ?==>? (Y,d) is complete

One might guess that it is not true. So would someone happen to know of a counterexample in which the set Y is closed in (X,d), but yet (Y,d) is *not* complete?

References:
1. Kubrusly(2011) Theorem 3.40 page 129:

2. Haaser(1991) 6.10 Proposition page 75:

Dan

On Saturday, February 2, 2013 12:52:55 AM UTC+8, 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).
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> >
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> >
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> >
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> > If (Y,d) is complete, then Y is closed with respect to d. That is,
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> >
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> >
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> > complete==>closed.
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> >
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> >
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> >
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> > Alternatively, if (Y,d) is complete, then Y contains all its limit
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> >
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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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> >
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> > is, does someone know of any example that demonstrates that
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> >
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> > closed --> complete
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> >
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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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> >
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> > guess that it is not true.
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> > Many thanks in advance,
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> >
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> > Dan
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> You need to understand that "closed" and "open" don't characterize topologies.
>
> Rather "X is open in Y" describes a relationship between X and Y.
>
> To say that a space is complete or compact or Hausdorff makes a statement about a topological space. To say that a space is "closed" (as in your statement "closed -> complete") doesn't really mean anything.
>
> To make progress replace "closed -> complete" by something more formal and rigorous and precise.
>
>
>
> Paul Epstein

Date Subject Author
2/1/13 Achimota
2/1/13 Paul
2/1/13 Paul
2/1/13 fom
2/1/13 fom
2/2/13 Shmuel (Seymour J.) Metz
2/3/13 fom
2/3/13 Shmuel (Seymour J.) Metz
2/2/13 Achimota
2/2/13 Butch Malahide
2/2/13 quasi
2/2/13 Butch Malahide
2/2/13 Achimota
2/2/13 quasi
2/3/13 Achimota
2/3/13 Paul
2/3/13 Achimota
2/1/13 Butch Malahide
2/1/13 J. Antonio Perez M.
2/1/13 William Hughes
2/2/13 J. Antonio Perez M.
2/1/13 Butch Malahide
2/1/13 William Elliot
2/2/13 Butch Malahide
2/2/13 William Elliot
2/2/13 Butch Malahide
2/2/13 Shmuel (Seymour J.) Metz