From: wdh@linus.mitre.org (Dale Hall)
Subject: Re: Is this thing a fractal?
Followup-To: sci.fractals
Summary: A definition of topological dimension, plus a good reference.
Keywords: dimension.
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Date: Thu, 5 Nov 1992 15:56:37 GMT

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?
>
>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.

	The standard (to me) definition of topological dimension
	involves the construction of polyhedra that approximate the
	space under consideration, and then taking the dimension of
	the spaces thus constructed:

	Let X be a topological space, and U an open cover of X.  Then
	we can define the "nerve" N(U) of U by:

		S_0 = 0-simplices of N(U) = elements of U [i.e. the
		open sets of the cover U]

		S_1 = 1-simplices of N(U) = pairs {A_0,A_1} from U
		with A_0 \intersect A_1 non-empty

		S_2 = 2-simplices of N(U) = triples {A_0,A_1,A_2} from
		U with A_0 \intersect A_1 \intersect A_2 non-empty

		...

		S_k = k-simplices of N(U) = (k+1)-tuples {A_0,...,A_k}
		from U with non-empty total intersection.

		...

	The face maps are defined in the obvious fashion, and one can
	of course have orientations if one uses ordered sets in the
	definition of the S_k.  One can (for X normal) also define a
	canonical map from X into the geometric realization of N(U),
	unique in the sense that any two such maps are contiguous
	(thus homotopic). 

	If V is a refinement of U, there is a canonical mapping of
	simplicial complexes N(V) --> N(U) defined on the 0-simplices
	and extending, so one obtains an inverse system of polyhedra,
	corresponding to ever-finer covers of X.  For "nice enough"
	spaces, the inverse limit converges, in the sense that from
	some stage on, the maps N(V') --> N(V) are essentially
	simplicial subdivisions (i.e., finer triangulations of the
	same underlying space).  Taking any of the representations
	beyond this point yields a simplicial complex with a
	well-defined dimension equal to the highest dimension of any
	simplex.  This is the so-called "covering dimension" of the
	space X.  For not-so-nice spaces, one obtains a pro-simplicial
	complex and hires a shape theorist. 

	Placed in layman's terms, for X to have covering dimension X
	means that any open covering U of X has a refinement U' for
	which at most (n+1) elements of U' intersect non-trivially. 
	You should note that, since we are counting things, only
	counting numbers will show up.  No fractional dimensions here.

	A good reference for dimension theory is "Dimension Theory" by
	Hurewicz and Wallman, published in 1948 by Princeton
	University Press. There are no doubt more recent references; I
	seem to recall a book by Nagata, but am not familiar with it.

	That's all I know.
						Dale.