Date: Feb 1, 2013 5:16 PM
Author: Butch Malahide
Subject: Re: looking for example of closed set that is *not* complete in a<br> metric space
On Feb 1, 10:37 am, "Daniel J. Greenhoe" <dgreen...@yahoo.com> wrote:

> 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? I don't know for sure that it is not true, but I might

> guess that it is not true.

Do you happen to know an example of a metric space which is not

complete? If so, let (X,d) be that metric space, and let Y = X.

If not, do you know an example of a metric space in which some subset

is not closed? In that case, let (W,d) be that metric space, let X be

a non-closed subset of W, and let Y = X. Then (X,d) is an incomplete

metric space, (Y,d) is a closed subspace of (X,d), and (Y,d) is not

complete.