Paul
Posts:
258
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 1:09 PM
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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.
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> Rather "X is open in Y" describes a relationship between X and Y.
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> 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.
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> To make progress replace "closed -> complete" by something more formal and rigorous and precise.
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> Paul Epstein
To clarify, you did attempt more precision by saying "closed with respect to d" but you're misusing/misunderstanding the concept of "closed" here, and you need to review your notes. "closed with respect to d" is not correct.
Paul
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