Toplologies of the Hilbert cube

From: Stephen J. Herschkorn (sjherschko_at_netscape.net)
Date: 03/29/05 

Date: Tue, 29 Mar 2005 03:30:03 -0500

I am reviewing general topology on my own, and, as a solution to an
exercise in Munkres, I came to the following conclusion:

The product, uniform, and ell-2 toplogy on the Hilbert cube are equal.
The box topology on the Hilbert cube is finer that this.

I am 99% sure of my proof, but I am looking for verification of this
result. 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)).
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

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