

November 27, 2008

O. Ravsky


TkachenkoTomita Group Example Construction

A topological group G is called a TkachenkoTomita if G is countably compact, abelian, torsionfree and contains no notrivial convergent sequence.
The first example of a TkachenkoTomita group was constructed by M. Tkachenko under the Continuum Hypothesis. Later, the Continuum Hypothesis was weakened to the Martin Axiom for σcentered posets by A. Tomita, for countable posets by P. Koszmider, A. Tomita and S. Watson, and finally to the existence continuum many incomparable selective ultrafilters by R. MadariagaGarcia and A. Tomita. Yet, no ZFCexample of a TkachenkoTomita group is known.
The aim of our meeting is to understand the construction by P. Koszmider, A. Tomita and S. Watson.



December 4, 2008,
December 11, 2008

O. Ravsky


On algebraically Baire topological groups

A topological group $G$ is algebraically Baire if $CN\not=G$ for each countable subset $C$ of $G$ and each nowhere dense subset $N$ of $G$. A topological group $G$ is locally algebraically Baire if $int (CN)=\emptyset$ for each countable subset $C$ of $G$ and each nowhere dense subset $N$ of $G$. Every Baire topological group is locally algebraically Baire and every locally algebraically Baire topological group is algebraically Baire. These facts suggest to consider the following questions.
Question 1. Is every locally algebraically Baire topological group a Baire group?
Question 2. Is every algebraically Baire topological group a locally algebraically Baire group?
It seems that we have obtained the following partial answers to these questions.
Answer 1. Every not precompact locally algebraically Baire topological group is a Baire group.
Answer 2. Every $\omega$precompact algebraically Baire topological group is a locally algebraically Baire group.



February 12, 2009

I. Guran, O. Gutik


Divertissement

Some open problems on the theory of topological semigroups were possed



February 19, 2009,
February 26, 2009,
March 5, 2009

I. Guran


Embedding of a cancellative topological semigroup into a topological group

The problem of an embedding of topological semigroups into topological groups will be discuss.



March 12, 2009

O. Ravsky


Characterizing meager paratopological groups

We are going to finish the proof of the next
Theorem. A topological group $G$ is meager if and only if there
is a nowhere dense subset $A\subset G$ and a countable subset
$C\subset G$ such that $C\cdot A=G$.
At the seminar we shall consider the case when the group is precompact.

