NetNews Usenet Archive 1992 #18/NN_1992_18.iso/spool/sci/math/10320

Internet Message Format | 1992-08-17 | 1.6 KB

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From: [e-mail] (Michael Weiss)
Newsgroups: sci.math
Subject: space-eating curves: correction
Date: 17 Aug 92 09:33:08
Organization: Thinking Machines Corporation, Cambridge MA, USA
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In-reply-to: [e-mail]'s message of 14 Aug 92 16:51:16


I wrote:

   Topologically, H [the Hilbert cube, I^{aleph_null}] is the product of
   countably many copies of I with the product topology.  Equivalently, for
   x in R^{aleph_null}, define ||x|| to be sqrt(sum (x_i)^2) (setting ||x||
   = infinity of course if the series diverges; otherwise this is the usual
   l^2 norm); then sets of the form N_e(x) = {y : ||y-x|| < e} form a
   neighborhood base (with e>0, of course.)

What was I thinking?  This "norm" gives a finer topology than the product
topology.  (A neighborhood base for H with the product topology consists of
products of the form {Product A_n | n=1 to infinity}, where all A_n are
subintervals of I open in I, and all but finitely many A_n equal I.  Here
I=[0,1].  No such product can be a subset of the set N_1(0) defined above,
for example.)

This fallacious remark was really parenthetical to the main point of my
post, but for the record, what I had in mind was the fact that H is
metrizable.  A correct formula for a suitable metric is:

    d(x,y) = sqrt {sum ((x_n - y_n)/n)^2 | n=1 to infinity}

where x = (x_1,x_2,...) and y = (y_1,y_2,...)