Preservation of metrizable absolute retracts 
and soft maps by covariant topological functors

T. Banakh

Assume that a covariant functor $F:{\mathcal SM}etr\to{\mathcal SM}etr$ on the category of separable metric spaces preserves embeddings, homotopies and compacta. It is proven that {(i)} if $F(K)\in \,\text{AR}\,$ for every cell $K=[0,1]^n$, $n\in\Bbb N$, then the functor $F$ preserves the class of separable absolute retracts; (ii) if $F(K)\in\text{ANR}$ for every finite simplicial complex $K$ then $F$ preserves the class of separable absolute neighborhood retracts having the homotopy type of a compact $\text{ANR}$; (iii) if $F(K)\in\text{ANR}$ for every countable locally finite simplicial complex $K$ then the functor $F$ preserves the class of separable absolute neighborhood retracts. If additionally, the functor $F$ preserves preimages then it preserves the class of soft maps between separable metric spaces if and only if the map $F(\operatorname{pr}_1):F(Q\times Q)\to F(Q)$ is soft (here $\operatorname{pr}_1:Q\times Q\to Q$ is the projection and $Q$ is the Hilbert cube).

1. Banakh T.O., Teleiko A.B. On independence of axioms of normal functor in the category of compacta, Preprint, 1993. (in Russian).

2. Basmanov V.N. Covariant functors, retracts and dimensionality // DAN SSSR. 1983. V.271. ¹ 5. P.1033–1036. (in Russian).

3. Basmanov V.N. Covariant topological functors and metrizable absolute retracts // Abstracts of VI Tiraspol Sympos. Gen. Top. Appl. – Kishinev, 1991. P.26–27. (in Russian).

4. Zarichnyi M.M. Preservation of ANR($\frak M$)-spaces and infinite-dimensional manifolds by certain covariant functors // DAN SSSR. 1983. V.271. ¹ 3. P.524--528. (in Russian).

5. Zarichnyi M.M. Characterization of noncompact AE($n$)-spaces // Ukr. mat. zh. 1992. V.44. ¹ 7. P.986--988. (in Russian).

6. Zarichnyi M.M., Fedorchuk V.V. Covariant functors in the categories of topological spaces // Itogi nauki i tekhn. Ser. Algebra. Topol. Geom. V.28. – M.: VINITI. 1990. P.47–95. (in Russian).

7. Fedorchuk V.V. Covariant functors in the category of compacta, absolute retracts and $Q$-manifolds // Uspekhi mat. nauk. 1981. V.36. ¹ 3. P.177--190. (in Russian).

8. Fedorchuk V.V. On some geometric properties of covariant functors // Uspekhi mat. nauk. 1984. V.39. ¹ 5. P.169–208. (in Russian).

9. Fedorchuk V.V. Soft maps, maltivalued retractions and functors // Uspekhi mat. nauk. 1986. ¹ 6. P.121–159. (in Russian).

10. Fedorchuk V.V. Probability measures in topoology // Uspekhi mat. nauk. 1991. V.46. ¹ 1. P.41–80. (in Russian).

11. Fedorchuk V.V., Filippov V.V. General topology. Fundamental constructions. –M.: Moscow Univ. Press, 1988. (in Russian).

12. Fedorchuk V.V., Chigogidze A.Ch. Extensors and soft maps. – M.: Moscow Univ. Press, 1987. (in Russian).

13. Chapman T. Lectures on Hilbert cube manifolds. – CBMS Reg. Conf. Ser. Math. No 28. Providence, 1976.

14. Chigogidze A.Ch. On extension of normal functors // Vestn. MGU. Ser. mat. 1984. ¹ 6. P.23–26. (in Russian).

15. Shchepin E.V. Functors and uncountable powers of compacta // Uspekhi mat. nauk. 1981. V.36. P.3-62. (in Russian).

16. Anderson R.D. Hilbert space is homeomorphic to countable infinite product of lines// Bull. Amer. Math. Soc. 1966. V.72. P.515–519.

17. Banakh T. Preservation of absolute (neighborhood) retracts and of soft maps by covariant toplogical functors// Proc. Ninth Annual Workshop in Geometric Topology, The Colorado College, June 11–13, 1992.– P.9–10.

18. Basmanov V.N. Covariant topological functors of finite degree and metrizable absolute retracts, preprint.

19. Bessaga C., Pe czyński A. Selected topics in infinite-dimensional topology. Warsaw: PWN, 1975.

20. Curtis D.W. Some theorems and examples on local equiconnectedness and its specializations// Fund. Math. 1971. V.72. P.101–113.

21. Fedor cuk V.V. Absolute retracts and some functors // General Topology and its Relations to Modern Analysis and Algebra V, Proc. Fifth Prague Topol. Symp. 1981, Berlin, 1982.– P.174–182.

22. Henderson D.W., West J.E. Triangulated infinite-dimensional manifolds// Bull. Amer. Math. Soc. 1970. V.76. P.655–660.

23. Hu S.-T. Theory of Retracts.– Detroit: Wayne State Univ. Press, 1965.

24. Nguen To Nhu. Investigating the ANR-property of metric spaces// Fund. Math. 1984. V.124, N.3. P.243-254.

25. Toruńczyk H. Absolute retracts as factors of normal linear spaces// Fund. Math. 1974. V.86, N.1. P.53–67.

26. Toru\'nczyk H. Concerning locally homotopy negligible sets and characterizing of $l_2$-manifolds// Fund. Math. 1978. V.101. P.93--110.

27. Toruńczyk H. Characterizing Hilbert space topology// Fund. Math. 1981. V.111, N.3. P.247–262. Department of Mathematics and Mechanics, Lviv University, Universytetska 1, 290602, Ukraine.

	Preservation of metrizable absolute retracts and soft maps by covariant topological functors
Author	T. Banakh
Abstract	Assume that a covariant functor $F:{\mathcal SM}etr\to{\mathcal SM}etr$ on the category of separable metric spaces preserves embeddings, homotopies and compacta. It is proven that {(i)} if $F(K)\in \,\text{AR}\,$ for every cell $K=[0,1]^n$, $n\in\Bbb N$, then the functor $F$ preserves the class of separable absolute retracts; (ii) if $F(K)\in\text{ANR}$ for every finite simplicial complex $K$ then $F$ preserves the class of separable absolute neighborhood retracts having the homotopy type of a compact $\text{ANR}$; (iii) if $F(K)\in\text{ANR}$ for every countable locally finite simplicial complex $K$ then the functor $F$ preserves the class of separable absolute neighborhood retracts. If additionally, the functor $F$ preserves preimages then it preserves the class of soft maps between separable metric spaces if and only if the map $F(\operatorname{pr}_1):F(Q\times Q)\to F(Q)$ is soft (here $\operatorname{pr}_1:Q\times Q\to Q$ is the projection and $Q$ is the Hilbert cube).
Keywords	covariant functor, separable metric spaces, embeddings, homotopies, compacta, finite simplicial complex, Hilbert cube
DOI	doi:10.30970/ms.4.1.95-104
Reference	1. Banakh T.O., Teleiko A.B. On independence of axioms of normal functor in the category of compacta, Preprint, 1993. (in Russian). 2. Basmanov V.N. Covariant functors, retracts and dimensionality // DAN SSSR. 1983. V.271. ¹ 5. P.1033–1036. (in Russian). 3. Basmanov V.N. Covariant topological functors and metrizable absolute retracts // Abstracts of VI Tiraspol Sympos. Gen. Top. Appl. – Kishinev, 1991. P.26–27. (in Russian). 4. Zarichnyi M.M. Preservation of ANR($\frak M$)-spaces and infinite-dimensional manifolds by certain covariant functors // DAN SSSR. 1983. V.271. ¹ 3. P.524--528. (in Russian). 5. Zarichnyi M.M. Characterization of noncompact AE($n$)-spaces // Ukr. mat. zh. 1992. V.44. ¹ 7. P.986--988. (in Russian). 6. Zarichnyi M.M., Fedorchuk V.V. Covariant functors in the categories of topological spaces // Itogi nauki i tekhn. Ser. Algebra. Topol. Geom. V.28. – M.: VINITI. 1990. P.47–95. (in Russian). 7. Fedorchuk V.V. Covariant functors in the category of compacta, absolute retracts and $Q$-manifolds // Uspekhi mat. nauk. 1981. V.36. ¹ 3. P.177--190. (in Russian). 8. Fedorchuk V.V. On some geometric properties of covariant functors // Uspekhi mat. nauk. 1984. V.39. ¹ 5. P.169–208. (in Russian). 9. Fedorchuk V.V. Soft maps, maltivalued retractions and functors // Uspekhi mat. nauk. 1986. ¹ 6. P.121–159. (in Russian). 10. Fedorchuk V.V. Probability measures in topoology // Uspekhi mat. nauk. 1991. V.46. ¹ 1. P.41–80. (in Russian). 11. Fedorchuk V.V., Filippov V.V. General topology. Fundamental constructions. –M.: Moscow Univ. Press, 1988. (in Russian). 12. Fedorchuk V.V., Chigogidze A.Ch. Extensors and soft maps. – M.: Moscow Univ. Press, 1987. (in Russian). 13. Chapman T. Lectures on Hilbert cube manifolds. – CBMS Reg. Conf. Ser. Math. No 28. Providence, 1976. 14. Chigogidze A.Ch. On extension of normal functors // Vestn. MGU. Ser. mat. 1984. ¹ 6. P.23–26. (in Russian). 15. Shchepin E.V. Functors and uncountable powers of compacta // Uspekhi mat. nauk. 1981. V.36. P.3-62. (in Russian). 16. Anderson R.D. Hilbert space is homeomorphic to countable infinite product of lines// Bull. Amer. Math. Soc. 1966. V.72. P.515–519. 17. Banakh T. Preservation of absolute (neighborhood) retracts and of soft maps by covariant toplogical functors// Proc. Ninth Annual Workshop in Geometric Topology, The Colorado College, June 11–13, 1992.– P.9–10. 18. Basmanov V.N. Covariant topological functors of finite degree and metrizable absolute retracts, preprint. 19. Bessaga C., Pe czyński A. Selected topics in infinite-dimensional topology. Warsaw: PWN, 1975. 20. Curtis D.W. Some theorems and examples on local equiconnectedness and its specializations// Fund. Math. 1971. V.72. P.101–113. 21. Fedor cuk V.V. Absolute retracts and some functors // General Topology and its Relations to Modern Analysis and Algebra V, Proc. Fifth Prague Topol. Symp. 1981, Berlin, 1982.– P.174–182. 22. Henderson D.W., West J.E. Triangulated infinite-dimensional manifolds// Bull. Amer. Math. Soc. 1970. V.76. P.655–660. 23. Hu S.-T. Theory of Retracts.– Detroit: Wayne State Univ. Press, 1965. 24. Nguen To Nhu. Investigating the ANR-property of metric spaces// Fund. Math. 1984. V.124, N.3. P.243-254. 25. Toruńczyk H. Absolute retracts as factors of normal linear spaces// Fund. Math. 1974. V.86, N.1. P.53–67. 26. Toru\'nczyk H. Concerning locally homotopy negligible sets and characterizing of $l_2$-manifolds// Fund. Math. 1978. V.101. P.93--110. 27. Toruńczyk H. Characterizing Hilbert space topology// Fund. Math. 1981. V.111, N.3. P.247–262. Department of Mathematics and Mechanics, Lviv University, Universytetska 1, 290602, Ukraine.
Pages	95-104
Volume	4
Issue	1
Year	1995
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Preservation of metrizable absolute retracts and soft maps by covariant topological functors