From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Measure 0 subfield of R Date: 4 Sep 2000 21:14:29 -0400 Newsgroups: sci.math Summary: [missing] In article <1eggnj0.7n7v3e1v12u2oN%jean_philippe.boucheron@cybercable.fr>, j-p boucheron wrote: [existence proof deleted] :Is "Hausdorff dimension zero" the same notion as : :"every point has a basis of open-closed sets" :(often called "zero-dimensional space")? : :If so, the usual Cantor set S is zero-dimensional, and the field it :generates is R. Hausdorff dimension of the middle-third Cantor set is ln(2)/ln(3), which is greater than 0.63 . A definition of Hausdorff dimension is found in many mathematically-oriented introductions to fractals. Cheers, ZVK(Slavek) ============================================================================== From: israel@math.ubc.ca (Robert Israel) Subject: Re: Measure 0 subfield of R Date: 5 Sep 2000 21:10:13 GMT Newsgroups: sci.math In article <1eggnj0.7n7v3e1v12u2oN%jean_philippe.boucheron@cybercable.fr>, j-p boucheron wrote: >Robert Israel wrote: >> Let S be an uncountable closed set of Hausdorff dimension 0 (these >> can be constructed by a modification of the Cantor middle-thirds >> construction), and K the field it generates. (I don't know why I said "closed": there's no need for S to be closed) >Is "Hausdorff dimension zero" the same notion as : >"every point has a basis of open-closed sets" >(often called "zero-dimensional space")? No. One characterization of a set S of Hausdorff dimension 0 in a metric space is the following: for any positive reals epsilon, delta and r, S can be covered by a sequence of closed sets S_j of diameter diam(S_j) < delta, such that sum_j diam(S_j)^r < epsilon. An example of a set of cardinality c and Hausdorff dimension 0 in the reals: the set of all x such that there are infinitely many pairs of integers p,q with |x - p/q| < 2^(-q). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2