Re: zero-dimensional spaces

From: Roger L. Bagula (rlbtftn_at_netscape.net)
Date: 08/23/04 

Date: Mon, 23 Aug 2004 16:49:31 GMT

Disconnected zero topological dimensional spaces are not necessarily
actual zero dimensioned.
Study of rational numbers from the fractal dimensional point of view
suggests that functional and Lyapunov exponent approaches give d<0.5
for the dimension of the rational number.
box counting give in two web sites: d=1 d=0.9
Hausdourff: d=0
Dr Edgar quotes d=1 in a sci.math post:
"the Bouligand dimension of
the set of rationals is 1."
It also implies that the irrational numbers with transcendentals and
without have fractal dimensions as well( both are disconnected sets
with the rational numbers between them).

It has to do with roughness theory as the basis of fractal
dimension.
> On 27 May 1999, Andrej Bauer wrote:
>
>> > I have encountered various questions about totally disconnected
>> topological spaces. Any classical or well-known texts that deal
>> with totally disconnected spaces in more than just a passing way,
>> and in sufficient generality (see below), would be very welcome.
>
> Engelking's `General Topology' contains a comprehensive discussion
> of `various kinds of disconnectedness' in Chapter 6.
>
>> > Definition 1:
>> A space X is *totally disconnected* if for every two points x and y
>> there exists a clopen (closed and open) set U that contains x and does
>> not contain y.
>
> This is the commonly accepted definition of total disconnectivity.
>
>> > Definition 2:
>> A space X is *totally disconnected* if it has a topological basis
>> consisting of clopen sets.
>
> This notion is commonly known as zero-dimensionality.
>
>> > Question 1: Is it true in general that Def. 1 implies Def. 2?
>> No, a famous example is the set of points in $\ell_2$ all of whose
> coordinates are rational --- this is due to Erd\H{o}s.
>
> Cheers,
> KP
>
> E-MAIL: K.P.Hart@twi.tudelft.nl PAPER: Department of Pure Mathematics
> PHONE: +31-15-2784572 TU Delft
> FAX: +31-15-2787245 Postbus 5031
> URL: http://aw.twi.tudelft.nl/~hart 2600 GA Delft
> the Netherlands
>
>
>

Respectfully, Roger L. Bagula
tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel:
619-5610814 :
URL : http://home.earthlink.net/~tftn
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