🚀 go-pugleaf

In article <1816@spam.ua.oz> ahanysz@spam.ua.oz (Alexander Hanysz) writes:
>
>Now I'm going to ask that eternally difficult question: WHY?  The
>Mandelbrot set is such a complicated object, how do we know that its
>boundary has topological dimension 1?

A proof was given earlier in this thread by Gerald Edgar.

>More fundamentally, does anyone know a nice _definition_ of topological
>dimension?  It's hard to argue when you don't know what the terms mean.
>All I've been told about topological dimension is "a line has dimension
>1, a surface has dimension 2, a volume..."  I have yet to see a formal
>definition.

I suspect that others have the same question so here goes:

   A set has topological dimension  0  if every point has arbitrarily
   small neighborhoods whose boundaries do not intersect the set.

   A set  S  has topological dimension  k  if each point in  S  has
   arbitrarily small neighborhoods whose boundaries meet  S  in a
   set of dimension  k-1 , and  k  is the least nonnegative integer
   for which this holds.

See, for example, Section 14.5 in Devaney's _A First Course in Chaotic
Dynamical Systems_.  (Does anyone know of a better reference?)

For instance, the rationals have topological dimension  0  since the
boundary of a disk with irrational radius fails to intersect the
rationals (that is, the sum of a rational and an irrational is
irrational).  The irrationals also have topological dimension  0 ,
I believe.  Suppose it were true that (1) if  A  is a subset of  X ,
then the  t-dim A <= t-dim X ; and (2) for any  A  in  R^n , t-dim A
= n  if and only if  A  contains a nonempty open subset of  R^n .
Then by (1) the irrationals have topological dimension no bigger than
1.  But by (2) they can not have topological dimension equal to  1
since they contain no open subset.  Hence, they have topological
dimension  0 .  (Can anyone think of a simpler proof?)

Here's another example.  Consider the fractal generated by the 
following geometric transformation:

                                  _____
                                 |     |
                                 |     |
     ---------------   -->   -----     -----

Each leg of this "hat" is one-third the length of the original line
segment.  Now the fractal dimension of the resulting image is
log 5 / log 3 , but the topological dimension is  1  since any disk
intersects the curve at a set of discrete points having topological
dimension  0 .  (There's a related exercise in Devaney that I can't
solve:  Can a fractal that is totally disconnected, with topological
dimension  0 , have a fractal dimension larger than  1 ?)

-- 
Tom Scavo
scavo@cie.uoregon.edu