Paul
Posts:
271
Registered:
7/12/10
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Re: looking for example of closed set that is *not* complete in a
metric space
Posted:
Feb 1, 2013 11:52 AM
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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
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> guess that it is not true.
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> Many thanks in advance,
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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
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