NetNews Usenet Archive 1992 #26/NN_1992_26.iso/spool/sci/fractals/280

Text File | 1992-11-04 | 3.5 KB | 88 lines

Newsgroups: sci.fractals
Path: sparky!uunet!think.com!linus!linus.mitre.org!linus!wdh
From: [e-mail] (Dale Hall)
Subject: Re: Is this thing a fractal?
Message-ID: <[e-mail]>
Followup-To: sci.fractals
Summary: A definition of topological dimension, plus a good reference.
Keywords: dimension.
Sender: Dale Hall
Nntp-Posting-Host: linus.mitre.org
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
References: <[e-mail]> <[e-mail]> <[e-mail]>
Date: Thu, 5 Nov 1992 15:56:37 GMT
Lines: 72

In article <[e-mail]> [e-mail] (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.