Axiomatic method of partitions in the theory

S. M. Ageev

Sbornik. Mathematics, Tome 198 (2007) no. 3, pp. 299-342

Résumé

The Nöbeling space $N_k^{2k+1}$, a $k$-dimensional analogue of the Hilbert space, is considered; this is a topologically complete separable (that is, Polish) $k$-dimensional absolute extensor in dimension $k$ (that is, $\mathrm{AE}(k)$) and a strongly $k$-universal space. The conjecture that the above-listed properties characterize the Nöbeling space $N_k^{2k+1}$ in an arbitrary finite dimension $k$ is proved. In the first part of the paper a full axiom system of the Nöbeling spaces is presented and the problem of the improvement of a partition connectivity is solved on its basis. Bibliography: 29 titles.

Bibliographie

[1] K. Menger, Dimensionstheorie, Teubner, Leipzig, 1928 | Zbl

[2] G. Nöbeling, “Über eine $n$-dimensionale Universalmenge im $\mathbb R^{2n+1}$”, Math. Ann., 104 (1931), 71–80 | DOI | MR | Zbl

[3] R. Geoghegan, R. Summerhill, “Pseudo-boundaries and pseudo-interiors in Euclidean spaces and topological manifolds”, Trans. Amer. Math. Soc., 194 (1974), 141–165 | DOI | MR | Zbl

[4] J. J. Dijkstra, J. van Mill, J. Mogilski, “Classification of finite-dimensional pseudo-boundaries and pseudo-interiors”, Trans. Amer. Math. Soc., 332:2 (1992), 693–709 | DOI | MR | Zbl

[5] A. Chigogidze, K. Kawamura, E. D. Tymchatyn, “Nöbeling spaces and the pseudo-interiors of Menger compacta”, Topology Appl., 68 (1996), 33–65 | DOI | MR | Zbl

[6] C. Bessaga, A. Pełczyński, Selected topics in infinite-dimensional topology, PWN, Warszawa, 1975 | MR | Zbl

[7] L. E. J. Brouwer, “On the structure of perfect sets of points”, Proc. Akad. Wet. Amsterdam, 12 (1910), 785–794

[8] R. D. Anderson, “A characterization of the universal curve and a proof of its homogeneity”, Ann. of Math. (2), 67:2 (1958), 313–324 | DOI | MR | Zbl

[9] H. Toruńczyk, “On CE-images of the Hilbert cube and characterization of $Q$-manifolds”, Fund. Math., 106:1 (1980), 31–40 | MR | Zbl

[10] M. Bestvina, Characterizing $k$-dimensional universal Menger compacta, Mem. Amer. Math. Soc., 71, No 380, 1988 | MR | Zbl

[11] P. S. Alexandroff, P. S. Urysohn, “Über nulldimensionale Punktmengen”, Math. Ann., 98:1 (1928), 89–106 | DOI | MR

[12] H. Toruńczyk, “Characterizing Hilbert space topology”, Fund. Math., 111:3 (1981), 247–262 | MR | Zbl

[13] K. Kawamura, M. Levin, E. D. Tymchatyn, “A characterization of 1-dimensional Nöbeling spaces”, Topology Proc., 22 (1997), 155–174 | MR | Zbl

[14] R. H. Bing, “Partitioning continuous curves”, Bull. Amer. Math. Soc., 58 (1952), 536–556 | DOI | MR | Zbl

[15] J. C. Mayer, L. G. Oversteegen, E. D. Tymchatyn, “The Menger curve. Characterization and extension of homeomorphisms of non-locally-separating closed subsets”, Dissertationes Math. (Rozprawy Mat.), 252 (1986), 45 pp. | MR | Zbl

[16] S. M. Ageev, Axiomatic method of partitions in the theory of Menger and Nöbeling spaces, Preprint, 2000; http://at.yorku.ca/v/a/a/a/87.htm

[17] A. S. Kechris, Classical descriptive set theory, Grad. Texts in Math., 156, Springer-Verlag, New York, 1995 | MR | Zbl

[18] K. Borsuk, Teoriya retraktov, Mir, M., 1971 ; K. Borsuk, Theory of retracts, PWN, Warszawa, 1967 | MR | MR | Zbl

[19] S. T. Hu, Theory of retracts, Wayne State Univ. Press, Detroit, 1965 | MR | Zbl

[20] J. Dugundji, E. Michael, “On local and uniformly local topological properties”, Proc. Amer. Math. Soc., 7:2 (1956), 304–307 | DOI | MR | Zbl

[21] E. Spener, Algebraicheskaya topologiya, Mir, M., 1971 ; E. H. Spanier, Algebraic topology, McGraw-Hill, New York–Toronto, ON–London, 1966 | MR | Zbl | MR | Zbl

[22] S. M. Ageev, S. A. Bogatyi, “O skleikakh nekotorykh klassov metricheskikh prostranstv”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 1994, no. 6, 19–23 | MR | Zbl

[23] P. Alexandroff, “Untersuchungen über Gestalt und Lage abgeschlossener Mengen beliebiger Dimension”, Ann. of Math. (2), 30:1/4 (1928–1929), 101–187 | DOI | MR | Zbl

[24] A. Kurosh, “Kombinatorischer Aufbau der bikompacten topologischen Räumen”, Compos. Math., 2 (1935), 471–476 | MR | Zbl

[25] P. S. Aleksandrov, B. A. Pasynkov, Vvedenie v teoriyu razmernosti, Nauka, M., 1973 | MR | Zbl

[26] R. Ekgelking, Obschaya topologiya, Nauka, M., 1986 ; R. Engelking, General topology, PWN, Warsaw, 1977 | MR | MR | Zbl

[27] M. Bestvina, J. Mogilskii, “Characterizing certain incomplete infinite-dimensional absolute retracts”, Michigan Math. J., 33:3 (1986), 291–313 | DOI | MR | Zbl

[28] J. Dugundji, “A duality property of nerves”, Fund. Math., 59 (1966), 213–219 | MR | Zbl

[29] K. Rurk, B. Sanderson, Vvedenie v kusochno lineinuyu topologiyu, Mir, M., 1974 ; C. P. Rourke, B. J. Sanderson, Introduction to piecewise-linear topology, Springer-Verlag, New York, 1972 | MR | MR | Zbl