Lebesgue's Density Theorem and definable selectors for ideals

We introduce a notion of density point and prove results analogous to Lebesgue's density theorem for various well-known ideals on Cantor space and Baire space. In fact, we isolate a class of ideals for which our results hold. In contrast to these results, we show that there is no reasonably definable selector that chooses representatives for the equivalence relation on the Borel sets of having countable symmetric difference. In other words, there is no notion of density which makes the ideal of countable sets satisfy an analogue to the density theorem. The proofs of the positive results use only elementary combinatorics of trees, while the negative results rely on forcing arguments.Comment: 28 pages; minor corrections and a new introductio

Generic trees

We continue the investigation of the Laver ideal ℓ0 and Miller ideal m 0 started in [GJSp] and [GRShSp]; these are the ideals on the Baire space associated with Laver forcing and Miller forcing. We solve several open problems from these papers. The main result is the construction of models for t < add(ℓ0), < add(m 0), where add denotes the additivity coefficient of an ideal. For this we construct amoeba forcings for these forcings which do not add Cohen reals. We show that = ω 2 implies add(m 0) ≤ . We show that , implies cov(ℓ0) ≤ +, cov(m 0) ≤ + respectively. Here cov denotes the covering coefficient. We also show that in the Cohen model cov(m 0) < holds. Finally we prove that Cohen forcing does not add a superperfect tree of Cohen real

Applications of the covering property axiom

The purpose of this work is two-fold. First, we present some consequences of the Covering Property Axiom CPA of Ciesielski and Pawlikowski which captures the combinatorial core of the Sacks\u27 model of the set theory. Second, we discuss the assumptions in the formulation of different versions of CPA.;As our first application of CPA we prove that under the version CPAgamecube of CPA there are uncountable strong gamma-sets on R . It is known that Martin\u27s Axiom (MA) implies the existence of a strong gamma-set on R . Our result is interesting since that CPAgamecube implies the negation of MA.;Next, we use the version CPAgameprism of CPA to construct some special ultrafilters on Q . An ultrafilter on Q is crowded provided it contains a filter basis consisting of perfect sets in Q . These ultrafilters have been constructed under various hypotheses. We study the properties of being P-point, Q-point, and o1-OK point and their negations; and prove under CPAgameprism the existence of an o1-generated crowded ultrafilter satisfying each consistent combination of these properties. We also refute an earlier claim by Ciesielski and Pawlikowski by proving under CPAgameprism that there are 2c -many crowded c -generated Q-points.;We also study various notions of density, central to the foundation of CPA and defined in the set of all perfect subsets of a Polish space X . These notions involve the concepts of perfect cube and iterated perfect set on Ca . If X is a Polish space, we say that F ⊆ Perf( X ) is alpha-cube (alpha-prism) dense provided for every continuous injection f : Ca → X there exists a perfect cube (iterated perfect set) C ⊆ Ca such that f[C] ∈ F .;We prove that for every alpha \u3c o1 and every Polish space X there exists a family F such that F is beta-prism dense for every beta \u3c alpha but &vbm0;X&bsol;&bigcup; F&vbm0;=c . Therefore, any attempt of strengthening of axiom CPAprism by replacing prism-density with any proper subclass of these densities leads to a false statement. The proof of this theorem is based in the following result: Any separately nowhere-constant function defined on a product of Polish spaces is one-to-one on some perfect cube

A Potpourri of Partition Properties

The cardinal characteristic inequality r <= hm3 is proved. Several partition relations for ordinals and one for countable scattered types are given. Moreover partition relations for lexicographically ordered sequences of zeros and ones are given in a no-choice context

Topological Groups: Yesterday, Today, Tomorrow

In 1900, David Hilbert asked whether each locally euclidean topological group admits a Lie group structure. This was the fifth of his famous 23 questions which foreshadowed much of the mathematical creativity of the twentieth century. It required half a century of effort by several generations of eminent mathematicians until it was settled in the affirmative. These efforts resulted over time in the Peter-Weyl Theorem, the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups, and finally the solution of Hilbert 5 and the structure theory of locally compact groups, through the combined work of Andrew Gleason, Kenkichi Iwasawa, Deane Montgomery, and Leon Zippin. For a presentation of Hilbert 5 see the 2014 book “Hilbert’s Fifth Problem and Related Topics” by the winner of a 2006 Fields Medal and 2014 Breakthrough Prize in Mathematics, Terence Tao. It is not possible to describe briefly the richness of the topological group theory and the many directions taken since Hilbert 5. The 900 page reference book in 2013 “The Structure of Compact Groups” by Karl H. Hofmann and Sidney A. Morris, deals with one aspect of compact group theory. There are several books on profinite groups including those written by John S. Wilson (1998) and by Luis Ribes and ‎Pavel Zalesskii (2012). The 2007 book “The Lie Theory of Connected Pro-Lie Groups” by Karl Hofmann and Sidney A. Morris, demonstrates how powerful Lie Theory is in exposing the structure of infinite-dimensional Lie groups. The study of free topological groups initiated by A.A. Markov, M.I. Graev and S. Kakutani, has resulted in a wealth of interesting results, in particular those of A.V. Arkhangelʹskiĭ and many of his former students who developed this topic and its relations with topology. The book “Topological Groups and Related Structures” by Alexander Arkhangelʹskii and Mikhail Tkachenko has a diverse content including much material on free topological groups. Compactness conditions in topological groups, especially pseudocompactness as exemplified in the many papers of W.W. Comfort, has been another direction which has proved very fruitful to the present day