
Re: Equivalent Spaces
Posted:
Aug 15, 2011 10:58 PM


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 Cauchy.
Dan

