From: pdt2@cornell.edu (Paul Thurston)
Newsgroups: sci.math
Subject: Re: Question on Tychonoff's Theorem
Date: Wed, 01 Feb 95 07:03:47 EST

In article <3gmoi2$ppc@news.csus.edu>, harrison@mercury.sfsu.edu says...
>
>Is it true that the dimension of the product of
>topological spaces is the product of their dimensions?
>Just wondering.
>--Craig Harrison
>
______________________________________________________________________

Hi Craig:

Let's specialize to separable metric spaces. It is a classical result 
from dimension theory that states

   1]      dim(X x Y) <= dimX + dimY.

There are examples (e.g. the Erdos space) for which equality fails in 1].

For the defintion of dimension and a delightful overview of this theory 
of see chapter 4 of "Infinite-Dimensional Topology: prerequisites and 
introduction" by J. Van Mill, North-Holland, 1989.

For other topics in dimension theory (like hausdorff dimension which is 
useful when studying fractals), see the classic text "Dimension Theory", 
by W. Hurewicz and H. Wallman, Princeton University Press, 1941. An 
appendix to this text also discuss some of the difficulties of extending 
dimension theory to more general spaces. 

Have fun in all those other dimensions!

Paul


-- 
******************************************************************
"You can't fool me Mr. Russell. I know it's turtles all the way down!"
******************************************************************
Paul Thurston
pdt2@cornell.edu

==============================================================================

From: edgar@math.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math,sci.fractals
Subject: dimension of product (was: Question on Tychonoff's Theorem
Date: Wed, 01 Feb 1995 08:58:49 -0500

> Let's specialize to separable metric spaces. It is a classical result 
> from dimension theory that states
> 
>    1]      dim(X x Y) <= dimX + dimY.
> 
> There are examples (e.g. the Erdos space) for which equality fails in 1].
> 

That would be the covering dimension, or one of the inductive dimensions
[they all agree for separable metric spaces].

The same inequality holds for the packing dimension:
          dim(X x Y) <= Dim X + Dim Y.
but the opposite holds for the Hausdorff dimension:
          dim(X x Y) >= dim X + dim Y.
A "fractal in the sense of Taylor" is a space X with
dim X = Dim X.  If X and Y are such spaces, then so is the
product X x Y and Dim(X x Y) = Dim X + Dim Y.
-- 
Gerald A. Edgar                  edgar@math.ohio-state.edu
Department of Mathematics
The Ohio State University
Columbus, OH 43210