An equivariant Hilbert cube generated
by 
the transversality mapping

A. Teleiko

The transversality mapping $\perp\colon GX\to GX$ defines an action of the group $\Bbb Z_2$ on the space $GX$ of inclusion hyperspaces of a compactum $X$. It is proved that $GX$ with this action is equimorphic to the equivariant Hilbert cube for a non-degenerate metrizable continuum $X$. Equivariant softness of the multiplication mapping $\mu\colon G\lambda X\to GX$ is proved for openly generated continua.

1. S.M. Ageev, Manifolds modeled on equivariant Hilbert cube // Mat. sb. 1994. V.185. ¹12. P.19–48. (Russian).

2. Fedorchuk V.V., Filippov V.V., General topology. Fundamental constructions, — Moscow: MGU. 1988. (Russian).

3. de Groot J. Supercompactness and superextensions // Proc. I Intern. Symp. Ext. Theory Topol. Struct. and its Appl. Berlin: Deutsch. Verlag. Wiss. 1967. P.89–90.

4. K. Kuratowski, Topologie I. — Warsaw-Wroc aw. 1948.

5. E.V. Moiseev, On spaces of closed growth and inclusion hyperspaces // Vestn. MGU, ser. I. Matem., Mech. 1988. ¹3. P.54–57. (Russian).

6. J. van Mill and M. van de Vel, On superextensions and hyperspaces. Topological Structure, II // Math. Centre Tracts, Math. Centrum Amsterdam, 1979. V.169. P.169–180.

7. T.N. Radul, Inclusion hyperspace monad and its algebras // Ukr. mat. zh. 1990. V.l42. P.806-811 (Russian).

8. E.V. Shchepin, Functors and uncountable powers of compacta // Uspekhi mat. nauk, 1981. V.36. P.3–62. (Russian).

9. H. Toruńczyk, On CE-images of the Hilbert cube and characterization of $Q$-manifolds // Fund. Math. 1980. V.106. P.31--40.

10. J. West, Infinite products which are Hilbert cubes // Trans. Amer. Math. Soc. 1970. V.150. P.1–25.

11. M.M. Zarichnyi, On covariant topological functors I // Q&A in General Topology. 1990. V.8. ¹2. P.317–360.

12. M.M. Zarichnyi, Absolute extensors and the geometry of multiplication of monads in the category of compacta // Math. USSR. Sbornik. 1993. V.74. ¹1. P.9–27.

	An equivariant Hilbert cube generated by the transversality mapping
Author	A. Teleiko Department of Mechanics and Mathematics, Lviv University
Abstract	The transversality mapping $\perp\colon GX\to GX$ defines an action of the group $\Bbb Z_2$ on the space $GX$ of inclusion hyperspaces of a compactum $X$. It is proved that $GX$ with this action is equimorphic to the equivariant Hilbert cube for a non-degenerate metrizable continuum $X$. Equivariant softness of the multiplication mapping $\mu\colon G\lambda X\to GX$ is proved for openly generated continua.
Keywords	transversality mapping, group action, inclusion hyperspaces, compactum, equivariant Hilbert cube, non-degenerate metrizable continuum
DOI	doi:10.30970/ms.7.2.205-210
Reference	1. S.M. Ageev, Manifolds modeled on equivariant Hilbert cube // Mat. sb. 1994. V.185. ¹12. P.19–48. (Russian). 2. Fedorchuk V.V., Filippov V.V., General topology. Fundamental constructions, — Moscow: MGU. 1988. (Russian). 3. de Groot J. Supercompactness and superextensions // Proc. I Intern. Symp. Ext. Theory Topol. Struct. and its Appl. Berlin: Deutsch. Verlag. Wiss. 1967. P.89–90. 4. K. Kuratowski, Topologie I. — Warsaw-Wroc aw. 1948. 5. E.V. Moiseev, On spaces of closed growth and inclusion hyperspaces // Vestn. MGU, ser. I. Matem., Mech. 1988. ¹3. P.54–57. (Russian). 6. J. van Mill and M. van de Vel, On superextensions and hyperspaces. Topological Structure, II // Math. Centre Tracts, Math. Centrum Amsterdam, 1979. V.169. P.169–180. 7. T.N. Radul, Inclusion hyperspace monad and its algebras // Ukr. mat. zh. 1990. V.l42. P.806-811 (Russian). 8. E.V. Shchepin, Functors and uncountable powers of compacta // Uspekhi mat. nauk, 1981. V.36. P.3–62. (Russian). 9. H. Toruńczyk, On CE-images of the Hilbert cube and characterization of $Q$-manifolds // Fund. Math. 1980. V.106. P.31--40. 10. J. West, Infinite products which are Hilbert cubes // Trans. Amer. Math. Soc. 1970. V.150. P.1–25. 11. M.M. Zarichnyi, On covariant topological functors I // Q&A in General Topology. 1990. V.8. ¹2. P.317–360. 12. M.M. Zarichnyi, Absolute extensors and the geometry of multiplication of monads in the category of compacta // Math. USSR. Sbornik. 1993. V.74. ¹1. P.9–27.
Pages	205-210
Volume	7
Issue	2
Year	1997
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

An equivariant Hilbert cube generated by the transversality mapping