Orthogonal retractions and $M$-equivalence

N. M. Pyrch

Two Tychonoff spaces are $M$-equivalent if their free topological groups are topologically isomorphic. We introduce the notion of orthogonal retracts in a Tychonoff space and as a~development of Okunev's construction show that every pair of orthogonal retracts leads to a~pair of $M$-equivalent spaces.

1. Aleksandryan R.A., Mirzahanyan E.A. General topology, Moscow, Vysshaya shkola, 1979.

2. Baars J., de Groot J., van Mill J. and Pelant J. An example of $l_{p}$-equivalent spaces which are not $l_{p}^{\ast }$-equivalent, Proc. Amer. Math. Soc., 119 (1993), no 3, 963--969.

3. Engelking R. General topology, Warszawa: PWN, 1977.

4. Guran I.Y., Zarichnyi M.M. Elements of the theory of topological groups, Kyiv, 1991.

5. Okunev O.G. A method for consructing examples of M-equivalent spaces, Topology Appl. 36 (1990), 157–171; Correction, Topology Appl. 49 (1993), 191–192.

6. Okunev O.G. M-equivalence of products, Trudy Mosk. Math. Obsch. 56 (1995), 192–205.

7. Pestov V.G. Universal arrows to forgetful functors from categories of topological algebra, Bull. Austr. Math. Soc. 48 (1993), 209–249.

8. Pyrch N., Zarichnyi M. On a generalization of Okunev's construction, Algebraic structures and their applications, Proceedings, 346–350.

9. Tkachenko M.G. Topological groups for topologists: Part 2, Bol.Soc.Mat. Mexicana(3) 6 (2000), 1–41.

10. Tkachuk V.V. Duality with respect to the $C_{p}$-functor and cardinal invariants of Souslin type, Mat. Zametki, 37 (1985), № 3, 441--451.

11. Tkachuk V.V. Homeomorphisms of free topological groups do not preserve compactness, Mat. Zametki, 42 (1987), № 3, 455–462.

	Orthogonal retractions and $M$-equivalence
Author	N. M. Pyrch Ivan Franko Lviv National University, Faculty of Mathematics and Mechanics
Abstract	Two Tychonoff spaces are $M$-equivalent if their free topological groups are topologically isomorphic. We introduce the notion of orthogonal retracts in a Tychonoff space and as a~development of Okunev's construction show that every pair of orthogonal retracts leads to a~pair of $M$-equivalent spaces.
Keywords	orthogonal retraction, M-equivalence, Tychonoff space
DOI	doi:10.30970/ms.20.2.151-161
Reference	1. Aleksandryan R.A., Mirzahanyan E.A. General topology, Moscow, Vysshaya shkola, 1979. 2. Baars J., de Groot J., van Mill J. and Pelant J. An example of $l_{p}$-equivalent spaces which are not $l_{p}^{\ast }$-equivalent, Proc. Amer. Math. Soc., 119 (1993), no 3, 963--969. 3. Engelking R. General topology, Warszawa: PWN, 1977. 4. Guran I.Y., Zarichnyi M.M. Elements of the theory of topological groups, Kyiv, 1991. 5. Okunev O.G. A method for consructing examples of M-equivalent spaces, Topology Appl. 36 (1990), 157–171; Correction, Topology Appl. 49 (1993), 191–192. 6. Okunev O.G. M-equivalence of products, Trudy Mosk. Math. Obsch. 56 (1995), 192–205. 7. Pestov V.G. Universal arrows to forgetful functors from categories of topological algebra, Bull. Austr. Math. Soc. 48 (1993), 209–249. 8. Pyrch N., Zarichnyi M. On a generalization of Okunev's construction, Algebraic structures and their applications, Proceedings, 346–350. 9. Tkachenko M.G. Topological groups for topologists: Part 2, Bol.Soc.Mat. Mexicana(3) 6 (2000), 1–41. 10. Tkachuk V.V. Duality with respect to the $C_{p}$-functor and cardinal invariants of Souslin type, Mat. Zametki, 37 (1985), № 3, 441--451. 11. Tkachuk V.V. Homeomorphisms of free topological groups do not preserve compactness, Mat. Zametki, 42 (1987), № 3, 455–462.
Pages	151-161
Volume	20
Issue	2
Year	2003
Journal	Matematychni Studii
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