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Topic: Equivalent Spaces
Replies: 10   Last Post: Aug 15, 2011 10:58 PM

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Dan Hoey

Posts: 172
Registered: 12/6/04
Re: Equivalent Spaces
Posted: Aug 15, 2011 10:58 PM
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On 8/12/2011 10:41 PM, William Elliot wrote:
> On Fri, 12 Aug 2011, Stephen J. Herschkorn wrote:
>> Stephen J. Herschkorn wrote:
>>> William Elliot wrote:

>>>> Let p,d be two topologically equivalent metrics for S.
>>>> If (S,d) is complete metric space, is (S,p) complete?

>>> Consider the metric p(x,y) = |1/x - 1/y| on the interval (0, 1).

>> Oops. Not quite right. Call that metric d, and put it on the interval
>> (0, 1].

> It seems S = ((0,1),d) is homemorphic to (1,oo) which isn't complete,
> by the isometry x -> 1/x. But then S wouldn't be complete.

Actually, S and (1,oo) are isomorphic (as metric spaces) under the
reciprocal map. So of course neither is complete.

Consider instead T=((0,1],d) which is homeomorphic to (0,1]. (0,1]
fails to be complete, because sequences {s_n} approaching zero do
not converge; these are the only counterexamples to completeness.
But T is complete because those sequences {s_n} are not Cauchy in
metric d. Alternatively, T is isomorphic to [1,oo), which is
complete; the sequences {1/s_n} approach oo, showing that they aren't


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