Re: Toplologies of the Hilbert cube
From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 03/29/05
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Date: Tue, 29 Mar 2005 07:41:44 -0600
On Tue, 29 Mar 2005 04:17:48 -0800, William Elliot <marsh@privacy.net>
wrote:
>On Tue, 29 Mar 2005, Stephen J. Herschkorn wrote:
>
>> The product, uniform, and ell-2 toplogy on the Hilbert cube are equal. The
>> box topology on the Hilbert cube is finer that this.
>>
>> Is my conclusion correct?
>> Details:
>> The uniform topology on R^N is that generated by the metric d(x, y) =
>> sup(n, min(|x_n - y_n|, 1)).
>
>This is not the produce topology for it gives you open set
>(-1/2,1/2)^N = B(0^N,1/2)
Nobody said that this _was_ the produce topology, nor the product
topology. The assertion is that it induces the same topology on
the Hilbert cube.
>> The box topology on R^N is that for which the collection of all infinite
>> products of open intervals is a basis.
>>
>> Let L the set of all square-summable sequences in R. L is a subset of
>> R^N. The ell-2 metric determines a topology on L.
>>
>> H = product (n in N, [0, 1/(n+1)]). H is called the Hilbert cube. H is a
>> subset of L. Thus, as a subspace, it can have any one of the four
>> topologies.
>>
>> --
>> Stephen J. Herschkorn sjherschko@netscape.net
>>
************************
David C. Ullrich
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