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From: colonel@gloria.UUCP (Col. G. L. Sicherman)
Newsgroups: net.math
Subject: Re: cardinality of the unit square & the unit line
Message-ID: <536@gloria.UUCP>
Date: Sun, 28-Apr-85 11:31:55 EDT
Article-I.D.: gloria.536
Posted: Sun Apr 28 11:31:55 1985
Date-Received: Wed, 1-May-85 06:08:58 EDT
References: <1767@decwrl.UUCP> <72@harvard.ARPA> <54@utastro.UUCP>
Organization: The Jack of Clubs Precision Instruments Co.
Lines: 19

["A true seaman is never lost!" --Water Polo]

> The real trick is mapping the unit line onto the unit square *continuously*.
> This was first done by Peano many years ago.  I don't think his example is
> one-to-one, however.

Of course not.  If it were one-to-one and continuous, it would be a
homeomorphism--the line and the square would be topologically the same.

The usual way of showing that A and B have the same number of points
is to map A onto B (_not_ necessarily 1-to-1) and then map B onto A
(again, not necessarily 1-to-1).  Constructing a 1-to-1 map is then
routine -- if you know the routine.  This can be done with the line
and the square, and it won't be continuous!

The trick of alternating digits works fine except at the end (0.9999...).
-- 
Col. G. L. Sicherman
...{rocksvax|decvax}!sunybcs!colonel