We consider the family {\mathcal{CS}} of symmetric Cantor subsets of {[0,1]} . Each set in {\mathcal{CS}} is uniquely determined by a sequence {a=(a_{n})} belonging to the Polish space {X\mathrel{\mathop{:}}=(0,1)^{\mathbb{N}}} equipped with probability product measure μ. This yields a one-to-one correspondence between sets in {\mathcal{CS}} and sequences in X. If {\mathcal{A}\subset\mathcal{CS}} , the corresponding subset of X is denoted by {\mathcal{A}^{\ast}} . We study the subfamilies {\mathcal{H}_{0}} , {\mathcal{SP}} and {\mathcal{M}} of {\mathcal{CS}} , consisting (respectively) of sets with Haudsdorff dimension 0, and of strongly porous and microscopic sets. We have {\mathcal{M}\subset\mathcal{H}_{0}\subset\mathcal{SP}} , and these inclusions are proper. We prove that the sets {\mathcal{M}^{\ast}} , {\mathcal{H}_{0}^{\ast}} , {\mathcal{SP}^{\ast}} are residual in X, and {\mu(\mathcal{H}_{0}^{\ast})=0} , {\mu(\mathcal{SP}^{\ast})=1} .
In the paper there are discussed separation axioms for different f -density topologies on the real line. It is proved that f -density topologies generated by functions f with lim inf x→0 + f (x)x > 0 are completely regular but f -density topologies generated by f such that lim inf x→0 + f (x) x = 0 are not regular. PreliminariesThrough the paper we shall use standard notation: R will be the set of real numbers, L the family of Lebesgue measurable subsets of R and |E| the Lebesgue measure of a measurable set E. We shall also write Eor equivalently if lim h→0 + |(x;x+h)\E| h = 0. In the same way we define a left-hand density point of E. We say that x is a density point of E if x is a right-hand density point and a left-hand density point of E. We will denote by Φ d (E) the set of all density points of E. From Lebesgue Density Theorem it follows that, for any measurable set E, E ∼ Φ d (E). It is well known [8, Theorem 2.2] that a family T d = {E ∈ L: E ⊂ Φ d (E)} forms a topology on the real line, called the density topology. It is also known (compare [8]) that• the density topology is essentially stronger than the natural topology on R;• an interior of a set E ⊂ R in the density topology consists of all points from E which are density points of a measurable kernel of E (so, for measurable E, int T d E = E ∩ Φ d (E)); • a set E ⊂ R is nowhere dense in the density topology if and only if |E| = 0; • T d is neither first countable, nor Lindelöf, nor separable;
MSC (2010) 03E15, 28A05, 03E17, 54H05We extend results of Elekes and Máthé on monotone Borel hulls to an abstract setting of measurable space with negligibles. This scheme yields the respective theorems in the case of category and in the cases associated with the Mendez σ-ideals on the plane.
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