Topological diagonalizations and Hausdorff dimension
Abstract
The Hausdorff dimension of a product 
can be strictly greater than that of 
, even when the Hausdorff dimension of 
is zero. But when is countable, the Hausdorff dimensions of and are the same. Diagonalizations of covers define a natural hierarchy of properties which are weaker than "being countable" and stronger than "having Hausdorff dimension zero". Fremlin asked whether it is enough for to have the strongest property in this hierarchy (namely, being a 
-set) in order to assure that the Hausdorff dimensions of and are the same.
We give a negative answer: Assuming the Continuum Hypothesis, there exists a-set 
and a set 
with Hausdorff dimension zero, such that the Hausdorff dimension of 
(a Lipschitz image of ) is maximal, that is, 
. However, we show that for the notion of a strong -set the answer is positive. Some related problems remain open.
can be strictly greater than that of , even when the Hausdorff dimension of is zero. But when is countable, the Hausdorff dimensions of and are the same. Diagonalizations of covers define a natural hierarchy of properties which are weaker than "being countable" and stronger than "having Hausdorff dimension zero". Fremlin asked whether it is enough for to have the strongest property in this hierarchy (namely, being a -set) in order to assure that the Hausdorff dimensions of and are the same. We give a negative answer: Assuming the Continuum Hypothesis, there exists a
-set and a set with Hausdorff dimension zero, such that the Hausdorff dimension of (a Lipschitz image of ) is maximal, that is, . However, we show that for the notion of a strong -set the answer is positive. Some related problems remain open.DOI Code:
10.1285/i15900932v22n2p83
Keywords:
Hausdorff dimension; Gerlits-Nagy $gamma$ property; Galvin-Miller strong $gamma$ property
Classification:
03E75; 37F20; 26A03
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