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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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 William Elliot Posts: 2,637 Registered: 1/8/12
closed but not complete
Posted: Feb 1, 2013 10:14 PM

On Fri, 1 Feb 2013, Daniel J. Greenhoe wrote:

A long title in ignarance of what he once learned in composition class.

> 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?

No. Assume K is a closed subset of the complete space (S,d).
Let s be Cauchy sequence into K. Then s is a Cauchy sequence
into S. Hence there's some x in S with s -> x. Since K is closed x in K.

Exercise. Show if S any topological space, K a closed subset of S,
and s a sequence into K that converges into x, then x is in K.

Conclusion. K subset complete S implies (K closed iff K complete).

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