Topological classification of zero-dimensional $\cal M_\omega$-groups

T.O.Banakh

A topological group $G$ is called an $\cal M_\omega$-group if it admits a countable cover $\cal K$ by closed metrizable subspaces of $G$ such that a subset $U$ of $G$ is open in $G$ if and only if $U\cap K$ is open in $K$ for every $K\in\cal K$. It is shown that any two non-metrizable uncountable separable zero-dimensional $\cal M_\omega$-groups are homeomorphic. Together with Zelenyuk's classification of countable $k_\omega$-groups this implies that the topology of a non-metrizable zero-dimensional $\cal M_\omega$-group $G$ is completely determined by its density and the compact scatteredness rank $r(G)$ which, by definition, is equal to the least upper bound of scatteredness indices of scattered compact subspaces of $G$.

1. Banakh Т. On topological groups containing a Fréchet-Urysohn fan Matem. Studii 9 (1998), № 2, 149–154.

2. F. van Engelen, Homogeneous zero-dimensional absolute Borel sets (CWI Tracts) North-Holland, Amsterdam, 1986.

3. Энгелькинг Р. Общая топология, М.: Мир, 1986.

4. Engelking R. Theory of dimensions, finite and infinite, Heldermann Verlag, Lemgo, 1995.

5. Kechris A. S. Classical descriptive set theory, Springer-Verlag, 1995.

6. Pollard J. On extending homeomorphisms on zero-dimensional spaces, Fund. Math. 67 (1970), 39–48.

7. Protasov I, Zelenyuk E. Topologies on groups determined by sequences, Matem. Studies Monograph Series, VNTL, Lviv, 1999.

8. Sakai K. On ${\Bbb R}^\infty$-manifolds and $Q^\infty$-manifolds,\/ Topol. Appl. 18 (1984), 69--79.

9. Зеленюк Е. Г. Топологии на группах, определяемые компактами, Maтем. студії 5 (1995), 5–16.

	Topological classification of zero-dimensional $\cal M_\omega$-groups
Author	T.O.Banakh tbanakh@franko.lviv.ua Faculty of Mechanics and Mathematics, Lviv National University, ,Universytetska 1, 79000, Lviv, Ukraine
Abstract	A topological group $G$ is called an $\cal M_\omega$-group if it admits a countable cover $\cal K$ by closed metrizable subspaces of $G$ such that a subset $U$ of $G$ is open in $G$ if and only if $U\cap K$ is open in $K$ for every $K\in\cal K$. It is shown that any two non-metrizable uncountable separable zero-dimensional $\cal M_\omega$-groups are homeomorphic. Together with Zelenyuk's classification of countable $k_\omega$-groups this implies that the topology of a non-metrizable zero-dimensional $\cal M_\omega$-group $G$ is completely determined by its density and the compact scatteredness rank $r(G)$ which, by definition, is equal to the least upper bound of scatteredness indices of scattered compact subspaces of $G$.
Keywords	topological group, countable cover K by closed metrizable subspaces, Zelenyuk's classification of countable kω-groups, topology determined by density, compact scatteredness rank , least upper bound of scatteredness indices, scattered compact subspaces
DOI	doi:10.30970/ms.15.1.109-112
Reference	1. Banakh Т. On topological groups containing a Fréchet-Urysohn fan Matem. Studii 9 (1998), № 2, 149–154. 2. F. van Engelen, Homogeneous zero-dimensional absolute Borel sets (CWI Tracts) North-Holland, Amsterdam, 1986. 3. Энгелькинг Р. Общая топология, М.: Мир, 1986. 4. Engelking R. Theory of dimensions, finite and infinite, Heldermann Verlag, Lemgo, 1995. 5. Kechris A. S. Classical descriptive set theory, Springer-Verlag, 1995. 6. Pollard J. On extending homeomorphisms on zero-dimensional spaces, Fund. Math. 67 (1970), 39–48. 7. Protasov I, Zelenyuk E. Topologies on groups determined by sequences, Matem. Studies Monograph Series, VNTL, Lviv, 1999. 8. Sakai K. On ${\Bbb R}^\infty$-manifolds and $Q^\infty$-manifolds,\/ Topol. Appl. 18 (1984), 69--79. 9. Зеленюк Е. Г. Топологии на группах, определяемые компактами, Maтем. студії 5 (1995), 5–16.
Pages	109-112
Volume	15
Issue	1
Year	2001
Journal	Matematychni Studii
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