Newsgroups: sci.math
Subject: [0,1] homeomorphic to most of [0,1]^2
From: wft@math.canterbury.ac.nz (Bill Taylor)
Date: 23 Aug 92 13:51:21 +1200
Distribution: world
Organization: Department of Mathematics, University of Canterbury

There was a flurry of concern in sci.math a week or two ago, as to whether
or not there was a bi-continuous bijection (i.e. a homeomorphism) between
[0,1] and [0,1]^2 . Several posters wrote in with their conviction that
such *did* exist, probably confusing vaguely remembered classic examples such
as Cantor's non-continuous bijection, and Peano's continuous many-1 mapping.

There were some excellent replies, ranging from the esoteric, to pointing
out the obvious such as that [0,1] could be disconnected by removing a
single point, whereas  [0,1]^2  couldn't, so they could not possibly be
homeomorphic. Hopefully they have killed off this potential piece of
mathematical folklore.


During the course of the discussion, I mentioned in passing that......

> there *is* a homeomorphism from the unit line to "most of" the unit square.
> i.e. so that the image in the unit square has measure arbitrarily close to 1 .

An "almost-homeomorphism", one might say, from [0,1] to [0,1]^2 .

I have been asked (by email) to substantiate this claim. Being unable to find a
reference for it, I have reconstructed and asciied-up a standard example as it
was shown to me long ago.  I thought I would post it here, in case there should
be some others who would like to see such an example.


Though it sounds amazing at first, the existence of an almost-homeomorphism
will come as no surprise to topologists, who know that there is almost no
connection between topology and measure theory. For instance there are
dense sets of measure zero (the rationals), and nowhere-dense sets of
measure  1-e , in the unit interval, for arbitrarily small positive  e .

These last could be called "thick Cantor sets". A thick Cantor set is
constructed by removing central intervals from [0,1] and its subsequent
sub-intervals, NOT all of length 1/3 of their intervals, as in the standard
Cantor set, but of proportional lengths  e, (e^2)/2, (e^3)/4  etc, so that
the measure of the remnant is  > 1-e-e^2-e^3-... > 1-2e . The intervals removed
can be open, leaving a closed Cantor-like set; or closed, leaving the
"quasi-interior" (i.e. no internal end-points) of one; having the same measure.

The almost-homeomorphism I display, will have as its range, a subset of [0,1]^2
consisting (mostly) of the cross-product with itself, of one of these
quasi-interior-of-thick-Cantor-sets.
So this range will have measure very close to 1.

Our first step is to create the set [0,1]^2 from which countably many
cross-shaped sections have been removed, but crosses of decreasingly small
measure. I believe this remnant set is sometimes called "Sierpinski dust".
                                                         ~~~~~~~~~~~~~~~
It is totally disconnected, nowhere dense, but of very large measure.
I illustrate it here....

   ------------------------------------------------------------------------
   |      ||      | |      ||      |      |      ||      | |      ||      |
   |  +   ||  +   | |  +   ||  +   |      |  +   ||  +   | |  +   ||  +   |
   |      ||      | |      ||      |      |      ||      | |      ||      |
   |======||======| |======||======|      |======||======| |======||======|
   |      ||      | |      ||      |      |      ||      | |      ||      |
   |  +   ||  +   | |  +   ||  +   |      |  +   ||  +   | |  +   ||  +   |
   |      ||      | |      ||      |      |      ||      | |      ||      |
   ---------------' `---------------      ---------------' `---------------
   |                               |      |                               |
   ---------------. .---------------      ---------------. .---------------
   |      ||      | |      ||      |      |      ||      | |      ||      |
   |  +   ||  +   | |  +   ||  +   |      |  +   ||  +   | |  +   ||  +   |
   |      ||      | |      ||      |      |      ||      | |      ||      |
   |======||======| |======||======|      |======||======| |======||======|
   |      ||      | |      ||      |      |      ||      | |      ||      |
   |  +   ||  +   | |  +   ||  +   |      |  +   ||  +   | |  +   ||  +   |
   |      ||      | |      ||      |      |      ||      | |      ||      |
   ---------------------------------      ---------------------------------
   |                                                                      |
   |                    area  e  removed in this central cross            |
   |                                                                      |
   ---------------------------------      ---------------------------------
   |      ||      | |      ||      |      |      ||      | |      ||      |
   |  +   ||  +   | |  +   ||  +   |      |  +   ||  +   | |  +   ||  +   |
   |      ||      | |      ||      |      |      ||      | |      ||      |
   |======||======| |======||======|      |======||======| |======||======|
   |      ||      | |      ||      |      |      ||      | |      ||      |
   |  +   ||  +   | |  +   ||  +   |      |  +   ||  +   | |  +   ||  +   |
   |      ||      | |      ||      |      |      ||      | |      ||      |
   ---------------' `---------------      ---------------' `---------------
   |                               |      |   area (e^2)/4 removed here   |
   ---------------. .---------------      ---------------. .---------------
   |      ||      | |      ||      |      |      ||      | |      ||      |
   |  +   ||  +   | |  +   ||  +   |      |  +   ||  +   | |  +   ||  +   |
   |      ||      | |      ||      |      |      ||      | |      ||      |
   |======||======| |======||======|      |======||======| |=(e^3)/16==removed
   |      ||      | |      ||      |      |      ||      | |      ||      |
   |  +   ||  +   | |  +   ||  +   |      |  +   ||  +   | |  +   || etc. |
   |      ||      | |      ||      |      |      ||      | |      ||      |
   ------------------------------------------------------------------------

We will draw a continuous non-self-intersecting image of [0,1] in the unit square,
that covers this Sierpinski dust.

First join the 4 large subsquares with straight line segments as shown...

   -------------------------------------------------------------------
   |                             |      |                            |
   |                             |      |      ***** *               |
   |            ******           |      |     *       *              |
   |           *      *          |      |    *         *             |
   |          *        *         |      |    *          *            |
   |         *          *        |      |    *           *           |
   |        *            *       |      |   *            *           |
   |        *             ********------****             *           |
   |        *                    |      |               *            |
   |         *                   |      |              *             |
   |          *                  |      |              *             |
   |           *                 |      |              *             |
   |            *                |      |              *             |
   |             *               |      |              *             |
   |             *               |      |              *             |
   --------------*----------------      ---------------*--------------
   |            1                                      |             |
   |                                                   |             |
   |            0                                     B|             |
   --------------*----------------      ---------------*--------------
   |             *               |      |              *             |
   |             *               |      |              *             |
   |             *               |      |               *            |
   |            *                |      |               *            |
   |           *                 |      |               *            |
   |          *                  |      |              *             |
   |          *                  |     A|             *              |
   |           *       ***********------********     *               |
   |            *     *          |      |       *****                |
   |             *****           |      |                            |
   |                             |      |                            |
   |                             |      |                            |
   |                             |      |                            |
   |                             |      |                            |
   |                             |      |                            |
   -------------------------------------------------------------------

The dotted lines show where further connections still have to be made.
"0" and "1" label the ends of our image-set. Note that the three solid lines
form a fixed part of the final curve; they will not be overwritten later
(unlike the Peano example).     Points "A" & "B" appear again below.

To define the curve inside each quarter, join up as shown for the magnified bottom
right quarter...   (again, the dotted sections to be filled later)  |
                                                                    |
                                                                    |
                                    B                               |
       -----------------------------*-------------------------------|
       |                         |  |  |                            |
       |                         |  |  |                            |
       |                         |  |  |                            |
       |                         | /   |                            |
       |                         |/    |                            |
       |              ************     **********                   |
       |             *           |    /|         *                  |
       |            *            |   / |          *                 |
       |           *             |  /  |           *                |
       |           *             |  |  |            *               |
       |           *             |  |  |             *              |
       ------------*-------------   |   --------------*-------------|
       |            \               |                 |             |
     A *----------\  \--------------/                 |             |
       |           \                                  |             |
       -------------*------------       --------------*-------------|
       |            *            |     |              *             |
       |            *            |     |              *             |
       |            *            |     |              *             |
       |            *            |     |              *             |
       |            *            |     |              *             |
       |             *************-----***************              |
       |                         |     |                            |
       |                         |     |                            |
       |                         |     |                            |
       |                         |     |                            |
       |                         |     |                            |
 -------------------------------------------------------------------|

The other three quarters are to have internal joins in a similar way.

Then make internal joins in the 16 size-(1/16) subsquares in a similar way
to this; and so on in reducing recursive fashion; countably many times.

The resulting curve is continuous and non-self-intersecting (i.e. 1-1),
though both of these facts need a little proof. The curve is made up of
countably many straight line sections, of total (areal) measure zero,
and an uncountable number of "connecting" limit points, the Sierpinski dust,
of measure close to 1.
                             Q.E.D.

-------------------------------------------------------------------------------
            Bill Taylor              wft@math.canterbury.ac.nz
-------------------------------------------------------------------------------
 Free will - the result of chaotic amplification of quantum events in the brain.
Galaxies - the result of chaotic amplification of quantum events in the big bang.
--------------------------------------------------------------------------------