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Set Theory, Topology and Banach Spaces (Second International Topology Conference in Kielce)
July 7-11, 2008
Institute of Mathematics, Jan Kochanowski University in Kielce; co-organized by Technical University of Lodz
Kielce, Poland

Organizers
Taras Banakh, Piotr Koszmider, Wieslaw Kubis

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Omega-1 limits of sets and covering of measure and category
by
Marcin Szyszkowski
Gdansk University, Gdansk
Coauthors: Andrzej Nowik, Tomasz Weiss

def. Suppose that F ⊆ P(R). We define
L(F) = ì
í
î
È
a < ℵ1 
Ç
b ≥ a 
 Ab : Ab ∈ F ü
ý
þ 		.
Each member of L(F) is called the w1 limit of a sequence of elements from F. We can summarise obvious inclusions by the diagram below, where "→" means inclusion " ⊂ ".


Picture Omitted

We point out that it is consistent with ZFC that 2w > ℵ1 and every subset of R is the w1 limit of a sequence of Gd sets in R. We prove also that assuming cov(null sets) > ℵ1, not every set in R is the w1 limit of a sequence of measurable sets. This solves two problems of T.Natkaniec and J.Wesoowska.

Date received: June 30, 2008

Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caxg-28.