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-------- Original Message --------
Subject: Re: Cantor set question -
Date: Tue, 07 Dec 2004 21:12:38 +0100
From: JEMebius <jemebius@xs4all.nl>
To: agapito6314@aol.com
Newsgroups: sci.math
References: <1102437980.557253.321310@f14g2000cwb.googlegroups.com>
The Cantor set is in the well-known way the intersection of an infinite
sequence {C0, C1, C2, ...} of sets C0 = [0, 1],
C1 = [0, 1/3] U [2/3, 1], etc. Each Cn is strictly contained in C(n-1).
Now think of the process of obtaining the successive set Cn from their
predecessors.
Sooner of later any (*) open segment around a real that contains a digit
1 in its triadic expansion (*) will drop out.
(*)...(*) I guess that this is what you mean by an open segment of the
form ( (3^k + 1)/3^m , (3^k + 2)/3^m ).
IHTH: Johan E. Mebius
=====
agapito6314@aol.com wrote:
[quote:84cf158187]Let P be the Cantor set, k and m any positive integers. How does one
prove that no (open) segment of the form
( (3^k + 1)/3^m , (3^k + 2)/3^m )
has a point in common with P? It appears as if segments of this form
are those middle thirds discarded in the construction of P, but how
does one prove formally?
Thanks for all help.
[/quote:84cf158187]
--
Respectfully, Roger L. Bagula
tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
alternative email: rlbtftn@netscape.net
URL : http://home.earthlink.net/~tftn |
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