Nonexistence of
absorbing set for a transfinite extension of covering dimension

T. Radul; M. Zarichnyi

R. Pol has shown that for every countable ordinal $\alpha$ there exists a universal space for separable metrizable spaces $X$ with $\operatorname{ind} X \le\alpha$. W. Olszewski has shown that for every countable limit ordinal $\lambda$ there is no universal space for separable metrizable space with $\operatorname{Ind} X\le\lambda$. We prove that for every countable limit ordinal there is no universal space for separable metrizable spaces with $\dim X\le\alpha$, where $\dim$ is a transfinite extension of covering dimension introduced by P. Borst. As an application, we show that there is no absorbing sets (in the sense of Bestvina and Mogilski) for the classes of spaces $X$ with $\dim X\le\alpha$ belonging to some absolute Borel class.

1. W. Hurewicz Ueber unendlich-dimensionale Punktmengen // Proc. Akad. Amsterdam – 1928 V. 31 p. 167–173

2. Yu.M. Smirnow On universal spaces for certain classes of infinite-dimensional spaces // Izv. Akad. Nauk SSSR, Ser. Math – 1959 V. 23 p. 185–196 Russian

3. R. Pol Countable dimensional universal set // Trans. Amer. Math. Soc. – 1986 V.297 p. 255–268

4. W. Olszewski Universal spaces in the theory of transfinite dimension // Fund. Math. – 1994 V. 144 p. 244-257

5. P. Borst Classification of weakly infinite-dimensional spaces // Fund. Math – 1988 V. 130 p. 1-25

6. R. Engelking Dimension theory ,addr Warszawa – 1978

7. M. Bestvina, J. Mogilski Characterizing certain incomplete infinite-dimensional absolute retracts // Michigan Math. J. – 1986 V. 33 p. 291–313

8. V. Chatyrko On the transfinite dimension $\dim$ // Q \& A in General Topology \toappear

9. V. Chatyrko Analogs of Cantors manifolds for transfinite dimensions // Matem. zametki – 1987 V. 42 p. 115–119 Russian

	Nonexistence of absorbing set for a transfinite extension of covering dimension
Author	T. Radul, M. Zarichnyi Lviv State University, Department of Mechanics and Mathematics, 290602 Lviv,Universytetska st.,1
Abstract	R. Pol has shown that for every countable ordinal $\alpha$ there exists a universal space for separable metrizable spaces $X$ with $\operatorname{ind} X \le\alpha$. W. Olszewski has shown that for every countable limit ordinal $\lambda$ there is no universal space for separable metrizable space with $\operatorname{Ind} X\le\lambda$. We prove that for every countable limit ordinal there is no universal space for separable metrizable spaces with $\dim X\le\alpha$, where $\dim$ is a transfinite extension of covering dimension introduced by P. Borst. As an application, we show that there is no absorbing sets (in the sense of Bestvina and Mogilski) for the classes of spaces $X$ with $\dim X\le\alpha$ belonging to some absolute Borel class.
Keywords	universal space, separable metrizable spaces, countable ordinal, covering dimension, transfinite extension of dimension, limit ordinal, absorbing sets, absolute Borel class
DOI	doi:10.30970/ms.9.1.94-98
Reference	1. W. Hurewicz Ueber unendlich-dimensionale Punktmengen // Proc. Akad. Amsterdam – 1928 V. 31 p. 167–173 2. Yu.M. Smirnow On universal spaces for certain classes of infinite-dimensional spaces // Izv. Akad. Nauk SSSR, Ser. Math – 1959 V. 23 p. 185–196 Russian 3. R. Pol Countable dimensional universal set // Trans. Amer. Math. Soc. – 1986 V.297 p. 255–268 4. W. Olszewski Universal spaces in the theory of transfinite dimension // Fund. Math. – 1994 V. 144 p. 244-257 5. P. Borst Classification of weakly infinite-dimensional spaces // Fund. Math – 1988 V. 130 p. 1-25 6. R. Engelking Dimension theory ,addr Warszawa – 1978 7. M. Bestvina, J. Mogilski Characterizing certain incomplete infinite-dimensional absolute retracts // Michigan Math. J. – 1986 V. 33 p. 291–313 8. V. Chatyrko On the transfinite dimension $\dim$ // Q \& A in General Topology \toappear 9. V. Chatyrko Analogs of Cantors manifolds for transfinite dimensions // Matem. zametki – 1987 V. 42 p. 115–119 Russian
Pages	94-98
Volume	9
Issue	1
Year	1998
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Nonexistence of absorbing set for a transfinite extension of covering dimension