Geodesic
On Luzin spaces
Sbornik. Mathematics, Tome 39 (1981) no. 3, pp. 405-415 Cet article a éte moissonné depuis la source Math-Net.Ru

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The main results of the paper are the following theorems: Theorem 1. {\it The following proposition is consistent with the system $ZFC$: $\mathscr {PMS}$. In a product of a family of not more than $2^\mathfrak c$ separable complete metric spaces without isolated points, there exists a dense Luzin subspace of cardinal $\mathfrak c$; if the family is uncountable, then every countable subset of the Luzin subspace is closed.} Theorem 2 [CH]. In a nondiscrete topological group every element of which has order 2, and whose space satisfies the Suslin conditions, has the Baire property and has $\pi$-weight not greater than $\mathfrak c$, there exists a dense Luzin subgroup. Theorem 3. The system $ZFC$ is consistent with the assertion that in any generalized Cantor discontinuum $D^m$ of infinite weight $m$ not greater than $2^\mathfrak c$, considered as a topological group, there exists a dense Luzin subgroup of cardinal $\mathfrak c$. Bibliography: 14 titles. 
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     author = {V. I. Malykhin},
     title = {On {Luzin} spaces},
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     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1981_39_3_a5/}
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V. I. Malykhin. On Luzin spaces. Sbornik. Mathematics, Tome 39 (1981) no. 3, pp. 405-415. http://geodesic.mathdoc.fr/item/SM_1981_39_3_a5/

[1] P. Dzh. Koen, Teoriya mnozhestv i kontinuum-gipoteza, izd-vo “Mir”, Moskva, 1969 | MR

[2] S. H. Hechler, “Short complete nested sequence in $\beta N/N$ and small maximal almostdisjont families”, Gen. Top. and Appl., 2:3 (1972), 139–149 | DOI | MR | Zbl

[3] S. H. Hechler, “Independence results concerning the number of nowhere dense sets necessary to cover the real line”, Acta math. Acad. sci. hung., 24:1–2 (1973), 27–32 | DOI | MR | Zbl

[4] A. Hainal and I. Juhasz, “A consistency result concerning hereditarily $\alpha$-separable spaces”, Indag. Math., 35:4 (1973), 301–307 | MR

[5] A. Hainal and I. Juhasz, “A consistency result concerning hereditarily $\alpha$-Lindelöf spaces”, Acta math. Acad. sci. hung., 24:3–4 (1973), 307–312 | MR

[6] K. Kunen, “Some points in $\beta N$”, Math. Proc. Camb. Phil. Soc., 80:6 (1976), 385–398 | DOI | MR | Zbl

[7] P. Fopenka and K. Hrbacek, “The consistency of some theorem of real numbers”, Bull. Pol. Acad. Sci., ser. Math., Astron., Phys., 15 (1967), 107–111 | MR

[8] E. K. van Douwen, F. D. Tall, W. A. R. Weiss, “Non-metrizable hereditarily Lindelof spaces with point-countable bases from $CH$”, Fundam. Math.

[9] K. Kuratovskii, Topologiya, t. 1, izd-vo “Mir”, Moskva, 1969 | MR

[10] G. P. Amirdzhanov, B. E. Shapirovskii, “O vsyudu plotnykh podmnozhestvakh topologicheskikh prostranstv”, DAN SSSR, 214:2 (1974), 249–252 | MR

[11] T. Iekh, Teoriya mnozhestv i metod forsinga, izd-vo “Mir”, Moskva, 1973

[12] Dzh. Shenfild, Matematicheskaya logika, izd-vo “Nauka”, Moskva, 1975 | MR | Zbl

[13] R. Engelking, Outline of general topology, North-Holland Publishing Company, Amsterdam, 1968. | MR | Zbl

[14] K. Kunen, “Luzin spaces”, Topology Proc, 1 (1976), 191–199 | MR | Zbl