Universal maps and absorbing sets in the classes of 
countable-dimensional spaces

M. Zarichnyi

It is proved that there exists a map $f:X\to Q$ of countable-dimensional (c.d.) space $X\in \mathcal A_2$ which is soft in the class of c.d. spaces (i.e. each partial section of $f$ parametrized by a closed subset of c.d. space can be extended over the whole space). The result is applied to constructing of absorbing sets in the classes of absolute Borelian and projective c.d. spaces. Some results are extended onto non-separable case and onto the case of pairs of spaces.

1. Aleksandrov P.S., Pasynkov B.A. Introduction to dimension theory, -- Nauka. Moscow. 1973. (in Russian).

2. Bestvina M., Mogilski J. Characterizing certain incomplete infinite-dimensional absolute retracts // Michigan Math. J. 33 (1986). P.291–313.

3. Bestvina M., Mogilski J. Linear maps do not preserve countable-dimensionality // Proc. Amer. Math. Soc. 93 (1985). P.661–666.

4. Cauty R. Ensembles absorbants pour les classes projectives // Fund. Math. 143 (1993). P.23–36.

5. Cauty R., Dobrowolski T., Applying coordinate products to the topological identification of normed spaces // Trans. Amer. Math. Soc. 337 (1993). P.625–649.

6. Chigogidze A.Ch. $n$-soft maps of $n$-dimensional spaces // Mat. Zametki. 46 (1989). P.88–95. (in Russian).

7. Chigogidze A.Ch., Valov V. Universal maps and surjective characterizations of completely metrizable $LC^n$-spaces // Proc. Amer. Math. Soc. 109 (1990). P.1125–1133.

8. Dobrowolski T., Mogilski J. Problems on topological classification of incomplete metric spaces, in: Open problems in topology, J. van Mill and G.M. Reeds (eds.) North Holland. Amsterdam. 1990. P.409–429.

9. Dranishnikov A.N. Absolute extensors in dimension $n$ and $n$-soft maps increasing dimension // Uspekhi Mat. Nauk. 39 (1984). P.55–95. (in Russian).

10. Engelking R., Pol E. Countable-dimensional spaces: a survey // Diss. Math. 216 (1983). P.5–41.

11. Katetov M. On dimension of non-separable spaces // Czechosl. Mat. Zh. 2(1952). P.333–368 (in Russian).

12. Kuratowski K. Topology, Vol. 1 -- Academic Press and PWN, 1966.

13. Nagata J. On the countable sum of zero-dimensional spaces // Fund. Math. 48 (1960). P.1–14.

14. Pol R. Countable-dimensional universal sets // Trans. Amer. Math. Soc. 297 (1986). P.255–268.

15. Shchepin E.V. Functors and uncountable powers of compacta // Uspekhi Mat. Nauk. 36 (1981). P.3–62. (in Russian).

16. Toruńczyk H. Concerning locally homotopy negligible sets and characterization of $l\_2$-manifolds // Fund. Math. 101 (1978). P.93-100.

17. Radul T. Absorbing sets for countable-dimensional spaces // Matem. Studiï. 1995. No 4. P.105–110.

18. Mogilski J. Universal finite-to-one map and universal countable dimensional spaces, Preprint, 1994.

	Universal maps and absorbing sets in the classes of countable-dimensional spaces
Author	M. Zarichnyi Department of Mathematics and Mechanics, Lviv University, Universytetska 1, 290602, Ukraine.
Abstract	It is proved that there exists a map $f:X\to Q$ of countable-dimensional (c.d.) space $X\in \mathcal A_2$ which is soft in the class of c.d. spaces (i.e. each partial section of $f$ parametrized by a closed subset of c.d. space can be extended over the whole space). The result is applied to constructing of absorbing sets in the classes of absolute Borelian and projective c.d. spaces. Some results are extended onto non-separable case and onto the case of pairs of spaces.
Keywords	countable-dimensional space, soft map, partial section, extension, absorbing sets, absolute Borelian spaces, projective spaces
DOI	doi:10.30970/ms.4.1.85-94
Reference	1. Aleksandrov P.S., Pasynkov B.A. Introduction to dimension theory, -- Nauka. Moscow. 1973. (in Russian). 2. Bestvina M., Mogilski J. Characterizing certain incomplete infinite-dimensional absolute retracts // Michigan Math. J. 33 (1986). P.291–313. 3. Bestvina M., Mogilski J. Linear maps do not preserve countable-dimensionality // Proc. Amer. Math. Soc. 93 (1985). P.661–666. 4. Cauty R. Ensembles absorbants pour les classes projectives // Fund. Math. 143 (1993). P.23–36. 5. Cauty R., Dobrowolski T., Applying coordinate products to the topological identification of normed spaces // Trans. Amer. Math. Soc. 337 (1993). P.625–649. 6. Chigogidze A.Ch. $n$-soft maps of $n$-dimensional spaces // Mat. Zametki. 46 (1989). P.88–95. (in Russian). 7. Chigogidze A.Ch., Valov V. Universal maps and surjective characterizations of completely metrizable $LC^n$-spaces // Proc. Amer. Math. Soc. 109 (1990). P.1125–1133. 8. Dobrowolski T., Mogilski J. Problems on topological classification of incomplete metric spaces, in: Open problems in topology, J. van Mill and G.M. Reeds (eds.) North Holland. Amsterdam. 1990. P.409–429. 9. Dranishnikov A.N. Absolute extensors in dimension $n$ and $n$-soft maps increasing dimension // Uspekhi Mat. Nauk. 39 (1984). P.55–95. (in Russian). 10. Engelking R., Pol E. Countable-dimensional spaces: a survey // Diss. Math. 216 (1983). P.5–41. 11. Katetov M. On dimension of non-separable spaces // Czechosl. Mat. Zh. 2(1952). P.333–368 (in Russian). 12. Kuratowski K. Topology, Vol. 1 -- Academic Press and PWN, 1966. 13. Nagata J. On the countable sum of zero-dimensional spaces // Fund. Math. 48 (1960). P.1–14. 14. Pol R. Countable-dimensional universal sets // Trans. Amer. Math. Soc. 297 (1986). P.255–268. 15. Shchepin E.V. Functors and uncountable powers of compacta // Uspekhi Mat. Nauk. 36 (1981). P.3–62. (in Russian). 16. Toruńczyk H. Concerning locally homotopy negligible sets and characterization of $l\_2$-manifolds // Fund. Math. 101 (1978). P.93-100. 17. Radul T. Absorbing sets for countable-dimensional spaces // Matem. Studiï. 1995. No 4. P.105–110. 18. Mogilski J. Universal finite-to-one map and universal countable dimensional spaces, Preprint, 1994.
Pages	85-94
Volume	4
Issue	1
Year	1995
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Universal maps and absorbing sets in the classes of countable-dimensional spaces