Universal maps of $k_\omega$-spaces

O. Ye. Shabat; M. M. Zarichnyi

We introduce counterparts of the spaces $\mathbb R^\infty$ and $Q^\infty$ in the classes of spaces that are countable direct limits of compacta of prescribed weights. The counterpart of the space $Q^\infty$ is a countable direct limit of Tychonov cubes, and that of $\mathbb R^\infty$ is a countable direct limit of Dranishnikov universal spaces. A universal map, which is a generalization of the universal map between $\mathbb R^\infty$ and $Q^\infty$ defined by the second-named author, between these spaces is constructed.

1. R.Heisey, Manifolds modeled on ${\Bbb R}^\infty$ or bounded weak-$^*$ topologies, Trans. Amer. Math. Soc. 206 (1975), 295--312.

2. T. Banakh, Parametric results for some classes of infinite-dimensional manifolds, Ukr. Mat. J. 43 № 6 (1991), 853–859.(Russian)

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

4. K. Sakai, On ${\Bbb R}^\infty$-manifolds and ${\Bbb Q}^\infty$-manifolds, II: Infinite deficiency, Tsukuba J. Math. 8 (1984), 101--118.

5. Vo Thang Liem, On infinite deficiency in $\mathbb R^\infty$-manifolds, Trans. Amer. Math. Soc. 288 (1985), 205--226.

6. K. Sakai, Combinatorial infinite-dimensional manifolds and $ {\Bbb R}^\infty$-manifolds. Topology Appl. 26 (1987), no. 1, 43--64.

7. K. Sakai, Each $ R^\infty$-manifold has a unique piecewise linear $ R^\infty$-structure. Proc. Amer. Math. Soc. 90 (1984), no. 4, 616--618.

8. M.M. Zarichnyi, On universal maps and spaces of probability measures with finite supports. Mathematical investigations, 78–82, 108, Pr. Lviv. Mat. Tov., 2, Lviv. Mat. Tov., Lviv, 1993.

9. M.M. Zarichnyi, Free topological groups of absolute neighborhood retracts and infinite-dimensional manifolds. Dokl. Akad. Nauk SSSR 266 (1982), no. 3, 541–544.(Russian)

10. M. M. Zarichnyi, Symmetric products that are finite-dimensional manifolds. Visnik Lviv. Derzh. Univ. Ser. Mekh.-Mat. (1985) No. 24, 65–69.(Ukrainian)

11. T. Banakh, K. Sakai, Free topological semilattices homeomorphic to $ {\Bbb R}^\infty$ or ${\Bbb Q}^\infty$. Topology Appl. 106 (2000), no. 2, 135--147.

12. T. Banakh, K. Sakai, Characterizations of $(\mathbb R^\infty,\sigma)$ or or $({\Bbb Q},\Sigma)$ -manifolds and their applications, Topol. Appl. 106 (2000), no. 2, 115-134.

13. M.M. Zarichnyi, Functors generated by universal maps of injective limits of sequences of Menger compacta, Matematika, V. 562, Riga, Latvian University, 95–102.

14. E. V. S cepin, Functors and uncountable degrees of compacta. Uspekhi Mat. Nauk, 36 (1981), no. 3, 3–62. (Russian)

15. M. M. Zarichny, Infinite-dimensional manifolds which arise from the direct limits of ANRs. Uspekhi Mat. Nauk 39 (1984), no. 2, 153–154. (Russian)

16. A. Teleiko, M. Zarichnyi, Categorical topology of compact Hausdorff spaces. – Lviv: VNTL Publishers, 1999. 263 pp.

17. A.N. Dranishnikov, Universal Menger compacta and universal mappings. Mat. Sb. (N.S.) 129(171) (1986), no. 1, 121–139. (Russian)

18. E. S cepin, Sur les applications continues des cubes de Tihonov. C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 4, 257–260.

19. T. Banakh, D. Repov s, On linear realizations and local self-similarity of the universal Zarichnyi map, Preprint.

20. M. Zarichnyi, Strongly countable-dimensional resolvents of sigma-compact groups. Fundam. Prikl. Mat. 4 (1998), № 1, 101–108.

	Universal maps of $k_\omega$-spaces
Author	O. Ye. Shabat, M. M. Zarichnyi Ivan Franko Lviv National University,Uniwersytet Rzeszowski
Abstract	We introduce counterparts of the spaces $\mathbb R^\infty$ and $Q^\infty$ in the classes of spaces that are countable direct limits of compacta of prescribed weights. The counterpart of the space $Q^\infty$ is a countable direct limit of Tychonov cubes, and that of $\mathbb R^\infty$ is a countable direct limit of Dranishnikov universal spaces. A universal map, which is a generalization of the universal map between $\mathbb R^\infty$ and $Q^\infty$ defined by the second-named author, between these spaces is constructed.
Keywords	countable direct limit, Tychonoff cube, universal map
DOI	doi:10.30970/ms.21.1.71-80
Reference	1. R.Heisey, Manifolds modeled on ${\Bbb R}^\infty$ or bounded weak-$^*$ topologies, Trans. Amer. Math. Soc. 206 (1975), 295--312. 2. T. Banakh, Parametric results for some classes of infinite-dimensional manifolds, Ukr. Mat. J. 43 № 6 (1991), 853–859.(Russian) 3. K. Sakai, On ${\Bbb R}^\infty$-manifolds and ${\Bbb Q}^\infty$-manifolds, Topol. Appl. 18 (1984), 69--79. 4. K. Sakai, On ${\Bbb R}^\infty$-manifolds and ${\Bbb Q}^\infty$-manifolds, II: Infinite deficiency, Tsukuba J. Math. 8 (1984), 101--118. 5. Vo Thang Liem, On infinite deficiency in $\mathbb R^\infty$-manifolds, Trans. Amer. Math. Soc. 288 (1985), 205--226. 6. K. Sakai, Combinatorial infinite-dimensional manifolds and $ {\Bbb R}^\infty$-manifolds. Topology Appl. 26 (1987), no. 1, 43--64. 7. K. Sakai, Each $ R^\infty$-manifold has a unique piecewise linear $ R^\infty$-structure. Proc. Amer. Math. Soc. 90 (1984), no. 4, 616--618. 8. M.M. Zarichnyi, On universal maps and spaces of probability measures with finite supports. Mathematical investigations, 78–82, 108, Pr. Lviv. Mat. Tov., 2, Lviv. Mat. Tov., Lviv, 1993. 9. M.M. Zarichnyi, Free topological groups of absolute neighborhood retracts and infinite-dimensional manifolds. Dokl. Akad. Nauk SSSR 266 (1982), no. 3, 541–544.(Russian) 10. M. M. Zarichnyi, Symmetric products that are finite-dimensional manifolds. Visnik Lviv. Derzh. Univ. Ser. Mekh.-Mat. (1985) No. 24, 65–69.(Ukrainian) 11. T. Banakh, K. Sakai, Free topological semilattices homeomorphic to $ {\Bbb R}^\infty$ or ${\Bbb Q}^\infty$. Topology Appl. 106 (2000), no. 2, 135--147. 12. T. Banakh, K. Sakai, Characterizations of $(\mathbb R^\infty,\sigma)$ or or $({\Bbb Q},\Sigma)$ -manifolds and their applications, Topol. Appl. 106 (2000), no. 2, 115-134. 13. M.M. Zarichnyi, Functors generated by universal maps of injective limits of sequences of Menger compacta, Matematika, V. 562, Riga, Latvian University, 95–102. 14. E. V. S cepin, Functors and uncountable degrees of compacta. Uspekhi Mat. Nauk, 36 (1981), no. 3, 3–62. (Russian) 15. M. M. Zarichny, Infinite-dimensional manifolds which arise from the direct limits of ANRs. Uspekhi Mat. Nauk 39 (1984), no. 2, 153–154. (Russian) 16. A. Teleiko, M. Zarichnyi, Categorical topology of compact Hausdorff spaces. – Lviv: VNTL Publishers, 1999. 263 pp. 17. A.N. Dranishnikov, Universal Menger compacta and universal mappings. Mat. Sb. (N.S.) 129(171) (1986), no. 1, 121–139. (Russian) 18. E. S cepin, Sur les applications continues des cubes de Tihonov. C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 4, 257–260. 19. T. Banakh, D. Repov s, On linear realizations and local self-similarity of the universal Zarichnyi map, Preprint. 20. M. Zarichnyi, Strongly countable-dimensional resolvents of sigma-compact groups. Fundam. Prikl. Mat. 4 (1998), № 1, 101–108.
Pages	71-80
Volume	21
Issue	1
Year	2004
Journal	Matematychni Studii
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