Monday, 13 September 2010

gn.general topology - A question about homeomorphic subsets of Hilbert space

Yes. All we need is to construct a continuous on H function f that is unbounded on C. After that, x,f(x)subsetHtimesmathbbC is closed, unbounded, and homeomorhic to C (in the obvious way) and HtimesmathbbC is isometric to H. Being closed and non-compact in an arbitrary separable metric space X implies the existence of such a function. The simplest construction is to take a countable open cover Uj of X that contains no finite subcover of C and to put f(x)=minfj(x) where fj(x)=fracjmin(1,operatornamedist(x,XsetminusUj)). Separability has actually nothing to do with it but in the non-separable case things become a bit more complicated.

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