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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

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 Paul Posts: 780 Registered: 7/12/10
Re: looking for example of closed set that is *not* complete in a
metric space

Posted: Feb 1, 2013 11:52 AM

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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> 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
>
> guess that it is not true.
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>
>
>
> Dan

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