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We prove that the lower density operator associated with the Baire category density points in the real line has Borel values of class Π 3 0 which is analogous to the measure case. We also introduce the notion of the Baire category density point of a subset with the Baire property of the Cantor space, and we prove that it generates a lower density operator with Borel values of class Π 3 0 . [ABSTRACT FROM AUTHOR]

A strong generalized topological space is an ordered pair X = ⟨ X , μ ⟩ such that X is a set and μ is a collection of subsets of X which covers X and is stable under arbitrary unions. A necessary and sufficient condition for a strong generalized topological space X to satisfy Urysohn’s lemma or its appropriate variant is shown in ZF . Notions of a U-normal and an effectively normal generalized topological space are introduced. It is observed that, in ZF , the Principle of Dependent Choices implies that every U-normal generalized topological space satisfies Urysohn’s lemma. It is shown that every effectively normal generalized topological space satisfies Csaszár’s modification of Urysohn’s lemma. In ZFA (also in ZF ), it is proved that the Axiom of Choice is equivalent to the statement “Every normal strong generalized topological space is effectively normal”. A ZF -example of a strong generalized topological normal space which satisfies the Tietze–Urysohn Extension Theorem and fails to satisfy Urysohn’s lemma is shown. Several intriguing open problems are posed. [ABSTRACT FROM AUTHOR]

We prove that the existence of a Borel lower density operator (a Borel lifting) with respect to the σ -ideal of countable sets, for an uncountable Polish space, is equivalent to CH. One of the implications is known (due to K. Musiał) and the remaining implication is derived from a general abstract result dealing with the negation of GCH. We observe that there is no lower density Borel operator with respect to the σ -ideal of countable sets, whose range is of bounded level in the Borel hierarchy. [ABSTRACT FROM AUTHOR]

James Earl Baumgartner (March 23, 1943-December 28, 2011) came of age mathematically during the emergence of forcing as a fundamental technique of set theory, and his seminal research changed the way set theory is done. He made fundamental contributions to the development of forcing, to our understanding of uncountable orders, to the partition calculus, and to large cardinals and their ideals. He promulgated the use of logic such as absoluteness and elementary submodels to solve problems in set theory, he applied his knowledge of set theory to a variety of areas in collaboration with other mathematicians, and he encouraged a community of mathematicians with engaging survey talks, enthusiastic discussions of open problems, and friendly mathematical conversations. [ABSTRACT FROM AUTHOR]