×

zbMATH — the first resource for mathematics

A method for constructing examples of M-equivalent spaces. (English) Zbl 0707.54007
Two Tychonoff spaces are called M-equivalent if their free topological groups in the sense of Markov are topologically isomorphic. Retractions \(r_ 1\) and \(r_ 2\) are parallel if \(r_ 1r_ 2=r_ 1\) and \(r_ 2r_ 1=r_ 2\). Images of the space X under parallel retractions are called parallel retracts of X. The author uses the notions of R-quotient mapping and space [S. M. Karnik and S. Willard, Can. Math. Bull. 25, 456-461 (1982; Zbl 0443.54020)]. He shows that if \(K_ 1\) and \(K_ 2\) are parallel retracts of a space X, then the R-quotient spaces \(X/K_ 1\) and \(X/K_ 2\) are M-equivalent. This allows to get a row of examples of pairs of M-equivalent spaces with different topological properties.
For instance Corollary 3.23: There exist M-equivalent spaces X and Y such that X is a first-countable locally compact space and Y is not a b-space and the tightness of Y is uncountable. In particular, the following properties are not preserved by M-equivalence: bisequentiality, the Fréchet-Urysohn property, sequentiality, k-property, countability of tightness.
Reviewer: L.B.Nachmanson

MSC:
54B15 Quotient spaces, decompositions in general topology
22A05 Structure of general topological groups
54C15 Retraction
54G20 Counterexamples in general topology
54D50 \(k\)-spaces
54D55 Sequential spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aarts, J.M.; Lutzer, D.J., Pseudocompactness and the product of Baire spaces, Pacific J. math., 48, 1-10, (1973) · Zbl 0238.54027
[2] Arhangel’skiǐ, A.V., Structure and classification of topological spaces and cardinal invariants, Uspekhi mat. nauk, Russian math. surveys, 33, 6, 33-95, (1978), English translation:
[3] Arhangel’skiǐ, A.V., O sootnošeniyah meždu invariantami topologičeskih gruppi ih podprostranstv, Uspekhi mat. nauk, 35, 3, 3-22, (1980)
[4] Arhangel’skiǐ, A.V., O lineynyh gomeomorfizmah prostranstv funktsii, Dokl. akad. nauk SSSR, 264, 6, 1289-1292, (1982)
[5] Arhangel’skiǐ, A.V., Functional tightness, Q-spaces and τ-embeddings, Comment. math. univ. carolin., 24, 1, 105-120, (1983) · Zbl 0528.54006
[6] Engelking, R., General topology, (1977), PWN Warsaw
[7] Glicksberg, I., Stone-čech compactifications of products, Trans. amer. math. soc., 90, 369-382, (1959) · Zbl 0089.38702
[8] Graev, M.I., Free topological groups, Izv. akad. nauk SSSR ser. mat., Amer. math. soc. transl., 8, 1, 279-324, (1962), English translation:
[9] Gul’ko, S.P.; Okunev, O.G., Lokal’naya kompaktnost’ i M-ekvivalentnost’, Voprosy geometrii i topologii, 14-23, (1986), Petrozavodsk
[10] Karnik, S.M.; Willard, S., Natural covers and R-quotient mappings, Canad. math. bull., 25, 4, 456-462, (1982) · Zbl 0443.54020
[11] Markov, A.A., On free topological groups, Izv. akad. nauk SSSR ser. mat., Amer. math. soc. transl., 8, 1, (1962), (in Russian)
[12] Okunev, O.G., Metod postroeniya primerov M-ekwiwalentnyh prostranstv, ()
[13] O.G. Okunev, Ob odnom sposobe postroeniya primerov M-ekwiwalentnyh prostranstv, VINITI 240-85 Dep.
[14] Okunev, O.G., O nesohranenii odnogo topologičeskogo svoystva otnošeniem M-ekwiwalentnosti, Nepreryvnye funktsii na topogičeskih prostranstvah, 123-125, (1986), Riga
[15] Okunev, O.G.; Shakhmatov, D.B., Svoistvo bera i lineinye izomorfizmy prostranstv nepreryvnyh funktsii, Topologičeskie struktury i ih otobraženiya, 89-92, (1987), Riga
[16] Pasynkov, B.A.; Vylov, V.M., O svobodnyh gruppah topologičeskih prostranstv, Comptes rendus acad. bulg. sci., 34, 8, 1049-1052, (1981)
[17] Pestov, V.G., The coincidence of the dimensions dim of l-equivalent topological spaces, Dokl. akad. nauk SSSR, Soviet math. dokl., 26, 2, 380-383, (1982), English translation: · Zbl 0518.54030
[18] Pestov, V.G., Nekotorye topologičeskie svoistva, sohraniaemye otnošeniem M-ekwiwalentnosti, Uspekhi mat. nauk, 39, 6, 203-204, (1984)
[19] Sierpiǹski, W., Un théoreme sur LES continus, Tôhoku math. J., 13, 300-303, (1918) · JFM 46.0299.03
[20] Tkačenko, M.G., On completeness of free abelian topological groups, Dokl. akad. nauk SSSR, Soviet math. dokl., 27, 2, 341-345, (1983), (in Russian) · Zbl 0521.22002
[21] Tkačuk, V.V., Ob odnom metode postroeniya primerov M-ekwiwalentnyh prostranstv, Uspekhi mat. nauk, 38, 6, 127-128, (1983)
[22] Tkačuk, V.V., Dvoystvennost’ otnositel’no funktora C_{p} i kardinal’nye invarianty tipa čisla suslina, Mat. zametki, 37, 3, 441-450, (1985)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.