• Tkachenko-Tomita Group Example Construction
  • A topological group G is called a Tkachenko-Tomita if G is countably compact, abelian, torsion-free and contains no no-trivial convergent sequence. The first example of a Tkachenko-Tomita group was constructed by M. Tkachenko under the Continuum Hypothesis. Later, the Continuum Hypothesis was weakened to the Martin Axiom for σ-centered 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. Madariaga-Garcia and A. Tomita. Yet, no ZFC-example of a Tkachenko-Tomita group is known. The aim of our meeting is to understand the construction by P. Koszmider, A. Tomita and S. Watson.

• 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.

• Divertissement
  • Some open problems on the theory of topological semigroups were possed

• Embedding of a cancellative topological semigroup into a topological group
  • The problem of an embedding of topological semigroups into topological groups will be discuss.

• 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.