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Covering dimension of topological products. (English. Russian original) Zbl 1153.54017

J. Math. Sci., New York 144, No. 3, 4031-4110 (2007); translation from Sovrem. Mat. Prilozh. 34 (2005).
This paper is dedicated to investigating the conditions under which the inequality
\[ \dim X \times Y \leq \dim X + \dim Y\tag{+} \]
holds. We already know that there are two compact metrizable 2-dimensional spaces such that their product is 3-dimensional from a result from L. S. Pontryagin. But generally we have that the inequality (+) is true. E.g., it holds for compact Hausdorff spaces and metrizable spaces. However we have an example of a product where one of the factors is metrizable and the other is Tychonoff. The question whether (+) holds for paracompact Hausdorff spaces is still open, also after this extensive research. (Question 3.25).
Using the concept of rectangularity the authors are able to close in on this open question. So they prove (Theorem 3:15) that (+) holds for paracompact Hausdorff spaces \(X\) and \(Y\) where \(X\) is a \(\Sigma(\theta)\) space and \(Y\) is a \(P(\theta)\) space. This product is also paracompact.
For conditions on one of the factors several results are strengthened. So if \(Y\) is \(\sigma\)-compact then (+) holds. Some results are proved when \(X\) is a metrizable space or the perfect image of a metrizable space.
The other objective is to construct spaces \(X\) and \(Y\) such that \(\dim X \times Y > \dim X + \dim Y\). Various examples have been constructed in the past. Using spaces of consistent families the first author managed to construct new examples of spaces \(X\) and \(Y\) such that \(\dim X \times Y > \dim X + \dim Y\). He constructed a perfectly normal product by the Juhász-Kunen-Rudin method modified by Wage and corrected by Przymusinski. He also constructed normal products by methods suggested by van Douwen and developed by Przymusinski.
The work is quite extensive and it gives an impression of the determination of the authors to strengthen results on the question of the product theorem beyond the field of metrizable spaces. It gives a good overview on the results proved on this issue and a basis for solving the main question whether for paracompact Hausdorff spaces the inequality (+) holds.

MSC:

54F45 Dimension theory in general topology
54B10 Product spaces in general topology
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References:

[1] P. S. Aleksandrov and B. A. Pasynkov, Introduction to Dimension Theory, Moscow, Nauka (1973).
[2] P. S. Aleksandrov and I. V. Proskuryakov, ”On reducible sets,” Izv. Akad. Nauk SSSR, Ser. Mat., 5, No. 3, 217–224 (1941).
[3] W. G. Bade, ”Two properties of the Sorgenfrey plane,” Pac. J. Math., 51, No. 2, 349–354 (1974). · Zbl 0308.54031
[4] R. L. Blair, ”Spaces in which special sets are z-embedded,” Can J. Math., 28, No. 4, 673–690 (1976). · Zbl 0359.54009 · doi:10.4153/CJM-1976-068-9
[5] M. G. Charalambous, ”The dimension of inverse limits,” Proc. Amer. Math. Soc., 58, No. 2, 289–295 (1976). · Zbl 0348.54029 · doi:10.1090/S0002-9939-1976-0410696-2
[6] M. G. Charalambous, ”An example concerning inverse limit sequences of normal spaces,” Proc. Amer. Math. Soc., 78, No. 4, 605–607 (1980). · Zbl 0451.54028 · doi:10.1090/S0002-9939-1980-0556641-1
[7] M. G. Charalambous, ”The dimension of inverse limit and N-compact spaces,” Proc. Amer. Math. Soc., 85, No. 4, 648–652 (1982). · Zbl 0489.54033
[8] M. G. Charalambous, ”Further theory and applications of covering dimension of uniform spaces,” Czech. Math. J., 41, No. 3, 378–394 (1991). · Zbl 0776.54024
[9] A. Ch. Chigogidze, ”On some questions in dimension theory,” Topology, Colloq. Math. Soc. Ja. Bolyai, 23, 273–286 (1980).
[10] A. Ch. Chigogidze, ”On the dimension of increments of Tychonoff spaces,” Fund. Math., 111, No. 1, 25–36 (1981). · Zbl 0378.54021
[11] A. Ch. Chigogidze, ”Zero-dimensional open mappings which increase dimension,” Comment. Math. Univ. Carolinae, 24, No. 4, 571–579 (1983). · Zbl 0538.54027
[12] C. H. Dowker, ”Inductive dimension of completely normal spaces,” Quart. J. Math. Oxford, Ser 2, 4, No. 16, 267–281 (1953). · Zbl 0052.39701 · doi:10.1093/qmath/4.1.267
[13] E. van Douwen, ”A technique for constructing honest locally compact submetrizable examples,” in: Topology Appl., 47, No. 3, 179–201 (1992). · Zbl 0770.54026 · doi:10.1016/0166-8641(92)90029-Y
[14] E. van Douwen, ”Mild infinite dimensionality of {\(\beta\)}X and {\(\beta\)}X / X for metrizable X,” Topology Appl., 51, No. 2, 93–108 (1993). · Zbl 0796.54040 · doi:10.1016/0166-8641(93)90143-2
[15] R. Engelking, ”On functions defined on Cartesian products,” Fund. Math., 59, No. 2, 221–231 (1966). · Zbl 0158.41203
[16] R. Engelking, General Topology, PWN, Warsaw (1977).
[17] R. Engelking, ”Theory of dimensions: Finite and infinite,” Sigma Ser. Pure Math., 10, Heldermann Verlag (1995). · Zbl 0872.54002
[18] R. Engelking and E. Pol, ”Countable-dimensional spaces,” Dissert. Math., 216, 1–45 (1983). · Zbl 0496.54032
[19] V. V. Fedorchuk, ”On the dimension of hereditarily normal spaces,” Proc. London Math. Soc., 36, No. 3, 163–175 (1978). · Zbl 0392.54020 · doi:10.1112/plms/s3-36.1.163
[20] V. V. Filippov, ”On the dimension of normal spaces,” Dokl. Akad. Nauk SSSR, 209, No. 4, 805–807 (1973). · Zbl 0292.54046
[21] V. V. Filippov, ”Abstracts of papers,” in: Int. Topological Conference, Moscow (1980), p. 90. · Zbl 0464.54042
[22] V. V. Filippov, ”On the dimension of topological products,” Fund. Math., 106, No. 3, 181–212 (1980). · Zbl 0464.54042
[23] V. V. Filippov, ”Normally posed subsets,” Tr. Mat. Inst. Steklova, 154, 239–251 (1983). · Zbl 0537.54028
[24] A. A. Fora, ”On the covering dimension of subspaces of products of Sorgenfrey lines,” Proc. Amer. Math. Soc., 72, No. 3, 601–606 (1978). · Zbl 0404.54029 · doi:10.1090/S0002-9939-1978-0509262-1
[25] A. A. Fora, ”The covering dimension of product of generalized Sorgenfrey lines,” Mat. Vestn., 5(18), No. 1, 33–41 (1981). · Zbl 0529.54033
[26] T. Hoshina and K. Morita, ”On rectangular products of topological spaces,” Topology Appl., 11, No. 1, 47–57 (1980). · Zbl 0431.54003 · doi:10.1016/0166-8641(80)90016-4
[27] M. Hušek, ”Mappings from products, Topological structures. II,” Math. Centre Tracts, 115, 131–145 (1979).
[28] T. Isiwata, ”Generalizations of M-spaces. I,” Proc. Japan. Acad., 45, 359–363 (1969). · Zbl 0181.50602 · doi:10.3792/pja/1195520758
[29] T. Isiwata, ”Generalizations of M-spaces. II,” Proc. Japan. Acad., 45, 364–367 (1969). · Zbl 0181.50603 · doi:10.3792/pja/1195520759
[30] T. Isiwata, ”Z-mappings and C-embeddings,” Proc. Japan. Acad., 45, 889–893 (1969). · Zbl 0209.27002 · doi:10.3792/pja/1195520555
[31] I. Juhász, K. Kunen, and M. E. Rudin, ”Two more hereditarily separable non-Lindelöf spaces,” Can. J. Math., 28, No. 5, 998–1005 (1976). · Zbl 0336.54040 · doi:10.4153/CJM-1976-098-8
[32] M. Katětov, ”A theorem on the Lebesgue dimension,” Časopis Pěst. Mat. Fys., 75, 79–87 (1950). · Zbl 0039.12303
[33] Y. Katuta, ”On the covering dimension of inverse limits,” Proc. Amer. Math. Soc., 84, No. 4, 588–592 (1982). · Zbl 0489.54032 · doi:10.1090/S0002-9939-1982-0643755-2
[34] B. S. Klebanov, ”On factorization of continuous mappings of product spaces,” in: IV Tiraspol Symp. on General Topology and Its Applications, Shtiintsa, Kishinev (1979), pp. 60–62. · Zbl 0434.62026
[35] B. S. Klebanov, Continuous images of topological products of metric spaces [in Russian], Thesis, Moscow (1980). · Zbl 0486.54007
[36] Y. Kodama, ”On subset theorems and the dimension of products,” Amer. J. Math., 91, No. 4, 486–498 (1969). · Zbl 0183.27702 · doi:10.2307/2373521
[37] K. L. Kozlov and B. A. Pasynkov, ”Dimension of subsets of topological products,” Proc. Second Soviet-Japan Symp. of Topology, Khabarovsk, 1989. Q.&A. in General Topology, 8, No. 1, 227–250 (1990).
[38] K. Kuratowski, Topology, Academic Press, New York-London; PWN, Warsaw (1966).
[39] K. Kuratowski and C. Ryll-Nardzewski, ”A general theorem on selectors,” Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 13, 397–403 (1965). · Zbl 0152.21403
[40] A. Lelek, ”Dimension inequalities for unions and mappings of separable metric spaces,” Colloq. Math., 23, No. 2, 69–91 (1971). · Zbl 0219.54032
[41] J. E. Mack and D. G. Jonson, ”The Dedekind completion of C(X),” Pac. J. Math., 20, No. 2, 231–243 (1967). · Zbl 0152.39802
[42] S. Mazurkiewicz, ”Sur les problems {\(\kappa\)} et {\(\lambda\)} de Urysohn,” Fund. Math., 10, 311–319 (1927). · JFM 53.0563.02
[43] E. Michael, ”The product of a normal space and a separable metric space need not be normal,” Bull. Amer. Math. Soc., 69, No. 3, 375–376 (1963). · Zbl 0114.38904 · doi:10.1090/S0002-9904-1963-10931-3
[44] J. van Mill, Infinite-Dimensional Topology, North-Holland Math. Library, 43, North-Holland, Amsterdam (1989).
[45] K. Morita, ”On the dimension of product spaces,” Amer. J. Math., 75, No. 2, 205–223 (1953). · Zbl 0053.12406 · doi:10.2307/2372449
[46] K. Morita, ”On the product of paracompact spaces,” Proc. Japan Acad., 39, No. 8, 559–563 (1963). · Zbl 0204.22702 · doi:10.3792/pja/1195522956
[47] K. Morita, ”Products of normal spaces with metric spaces,” Math. Ann., 154, No. 4, 365–382 (1964). · Zbl 0117.39803 · doi:10.1007/BF01362570
[48] K. Morita, ”On the dimension of the product of Tychonoff spaces,” Gen. Topology Appl., 3, No. 2, 125–133 (1973). · Zbl 0258.54034 · doi:10.1016/0016-660X(73)90014-7
[49] K. Morita, ”Čech cohomology and covering dimension for topological spaces,” Fund. Math., 87, No. 1, 31–52 (1975). · Zbl 0336.55003
[50] K. Morita, ”On the dimension of the product of topological spaces,” Tsukuba J. Math., 1, 1–6 (1977). · Zbl 0403.54021
[51] K. Morita, ”Dimension of general topological spaces,” in: Surveys in General Topology, Academic Press, New York-London-Toronto (1980), pp. 297–336.
[52] S. Mrówka, ”Recent results on E-compact spaces and structures of continuous functions,” in: Proc. Univ. of Oklahoma Topology Conf., 1972, Univ. of Oklahoma, Norman (1972), pp. 168–221.
[53] K. Nagami, ”Finite-to-one closed mappings and dimension II,” Proc. Japan Acad., 35, 437–439 (1959). · Zbl 0151.30404 · doi:10.3792/pja/1195524246
[54] K. Nagami, ”{\(\Sigma\)}-spaces,” Fund. Math., 65, No. 2, 169–192 (1969).
[55] K. Nagami, ”A note on the large inductive dimension of totally normal spaces,” J. Math. Soc. Japan, 21, No. 2, 282–290 (1969). · Zbl 0175.19901 · doi:10.2969/jmsj/02120282
[56] K. Nagami, ”Countable paracompactness of inverse limits and products,” Fund. Math., 73, No. 3, 261–270 (1972). · Zbl 0226.54005
[57] K. Nagami, ”Dimension of non-normal spaces,” Fund. Math., 109, No. 2, 113–121 (1980). · Zbl 0456.54026
[58] J. Nagata, ”Product theorems in dimension theory,” Bull. Acad. Polon. Sci. Ser. Math. Astronom. Phys., 15, No. 7, 439–448 (1967). · Zbl 0164.23502
[59] J. Nagata and F. Siwiec, ”A note on nets and metrization,” Proc. Japan Acad., 44, 623–627 (1968). · Zbl 0181.25902 · doi:10.3792/pja/1195521079
[60] T. Nishiura, ”A subset theorem in dimension theory,” Fund. Math., 95, No. 2, 105–109 (1977). · Zbl 0354.54021
[61] P. Nyikos, ”The Sorgenfrey plane in dimension theory,” Fund. Math. 79, No. 2, 131–139 (1973). · Zbl 0264.54028
[62] A. V. Odinokov, ”On the dimension and rectangularity of product spaces with a Lashnev factor,” Vestn. Mosk. Univ., Ser. Mat. Mekh., 1, 59–62 (1999).
[63] H. Ohta, ”On normal nonrectangular products,” Quart. J. Math. Oxford., Ser. 2, 32, No. 127, 339–344 (1981). · Zbl 0459.54006 · doi:10.1093/qmath/32.3.339
[64] H. Ohta, ”Extensions of zero-sets in the products of topological spaces,” Topology Appl., 35, No. 1, 21–39 (1990). · Zbl 0776.54016 · doi:10.1016/0166-8641(90)90118-L
[65] A. Okuyama, ”Some generalizations of metric spaces, their metrization theorems and product theorems,” Sci. Rep. Tokyo Kyoiku Daigaku Sec. A, 9, 236–254 (1967). · Zbl 0153.52404
[66] B. A. Pasynkov, ”On universal compacta of given weight and given dimension,” Dokl. Akad. Nauk SSSR, 154, No. 5, 1042–1043 (1964). · Zbl 0197.48601
[67] B. A. Pasynkov, ”A class of mappings and the dimension of normal spaces,” Sib. Mat. Zh., 5, No. 2, 356–376 (1964). · Zbl 0124.15903
[68] B. A. Pasynkov, ”Partial topological products,” Tr. Mosk. Mat. Obshch., 13, 136–245 (1965). · Zbl 0162.26402
[69] B. A. Pasynkov, ”On universal spaces,” Fund. Math., 60, No. 3, 285–308 (1968).
[70] B. A. Pasynkov, ”On the dimension of products of normal spaces,” Dokl. Akad. Nauk SSSR, 209, No. 4, 792–794 (1973). · Zbl 0287.54036
[71] B. A. Pasynkov, ”The dimension of rectangular products,” Dokl. Akad. Nauk SSSR, 221, No. 2, 291–294 (1975). · Zbl 0334.54024
[72] B. A. Pasynkov, ”On the dimension of topological products and limits of inverse systems,” Dokl. Akad. Nauk SSSR, 254, No. 6, 1332–1336 (1980).
[73] B. A. Pasynkov, ”Factorization theorems in dimension theory,” Usp. Mat. Nauk, 36, No. 3, 147–175 (1981). · Zbl 0477.54021
[74] B. A. Pasynkov, ”On monotonicity of dimension,” Dokl. Akad. Nauk SSSR, 267, No. 3, 548–552 (1982). · Zbl 0561.54028
[75] B. A. Pasynkov, ”On dimension theory. Aspects of topology,” London Math. Soc. Lect. Note Ser., 93, 227–250, Cambridge Univ. Press, Cambridge (1985).
[76] B. A. Pasynkov, ”A factorization theorem for the dimension Ä,” in: Geometry of Immersed Manifolds [in Russian], Moscow (1986), pp. 70–75.
[77] B. A. Pasynkov, ”Monotonicity of dimension and dimension-increasing open mappings,” Tr. Mat. Inst. Steklova, 247, 202–213 (2004).
[78] A. R. Pears and J. Mack, ”Closed covers, dimension and quasi-ordered spaces,” Proc. London Math. Soc., 29, No. 3, 289–316 (1974). · Zbl 0319.54026 · doi:10.1112/plms/s3-29.2.289
[79] E. Pol, ”On the dimension of the product of metrizable spaces,” Bull. Acad. Polon. Sci., 26, No. 6, 525–534 (1978). · Zbl 0398.54023
[80] T. V. Proselkova, ”On a relationship between various posedness types of spaces,” in: General Topology: Spaces, Maps, and Functors [in Russian], Moscow (1992), pp. 111–125. · Zbl 0878.54020
[81] T. Przymusiński, ”On the notion of n-cardinality,” Proc. Amer. Math. Soc., 69, No. 2, 333–338 (1978). · Zbl 0418.54010
[82] T. Przymusiński, ”On the dimension of normal spaces and an example of M. Wage,” Proc. Amer. Math. Soc., 76, No. 2, 315–321 (1979).
[83] T. Przymusiński, ”Normality and paracompactness in finite and countable Cartesian products,” Fund. Math., 105, No. 2, 87–104 (1980). · Zbl 0438.54021
[84] T. Przymusiński, ”Product spaces,” in: Surveys in General Topology (G. M. Reed, ed.), Academic Press, New York (1980), pp. 399–429.
[85] T. Przymusiński, ”Products of perfectly normal spaces,” Fund. Math., 108, No. 2, 129–136 (1980). · Zbl 0438.54022
[86] T. Przymusiński, ”A solution of a problem of E. Michael,” Pac. J. Math., 114, No. 1, 235–242 (1984). · Zbl 0558.54006
[87] L. R. Rubin, R. M. Schori, and J. J. Walsh, ”New dimension-theory techniques for constructing in.nite-dimensional examples,” Gen. Topology Appl., 10, No. 1, 93–102 (1979). · Zbl 0413.54042 · doi:10.1016/0016-660X(79)90031-X
[88] M. E. Rudin and M. Starbird, ”Products with a metric factor,” Gen. Topology Appl., 5, 235–248 (1975). · Zbl 0305.54010 · doi:10.1016/0016-660X(75)90023-9
[89] E. V. Shchepin, ”Real-valued functions and canonical sets in product spaces and topological groups,” Usp. Mat. Nauk, 31, No. 6, 17–27 (1976). · Zbl 0366.54005
[90] E. V. Shchepin, ”On topological products, groups, and a new class of spaces more general than the metric spaces,” Dokl. Akad Nauk SSSR, 226, No. 3, 527–529 (1976). · Zbl 0338.54022
[91] H. Tamano, ”A note on the pseudocompactness of product of two spaces,” Mem. Coll. Sci. Univ. Kyoto Ser. A Math., 33, 225–230 (1960). · Zbl 0101.15202
[92] K. Tamano, ”A note on E. Michael’s example and rectangular products,” J. Math. Soc. Japan, 34, No. 2, 187–190 (1982). · Zbl 0475.54004 · doi:10.2969/jmsj/03420187
[93] J. Terasawa, ”On the zero-dimensionality of some non-normal product spaces,” Sci. Rep. Tokyo Kyoiku Daigaku, Sec. A, 11, 167–174 (1972). · Zbl 0253.54032
[94] K. Tsuda, ”Some examples concerning the dimension of product spaces,” Math. Japon., 27, No. 2, 177–195 (1982). · Zbl 0475.54024
[95] K. Tsuda, ”An n-dimensional version of Wage’s example, Colloq. Math., 49, No. 1, 15–19 (1984). · Zbl 0525.54027
[96] K. Tsuda, ”A Wage-type example with a pseudocompact factor,” Topology Appl., 20, No. 2, 191–200 (1985). · Zbl 0562.54049 · doi:10.1016/0166-8641(85)90079-3
[97] K. Tsuda, Dimension theory of general spaces, Thesis, Univ. Tsukuba (1985).
[98] M. Wage, The dimension of product spaces, Preprint, Yale Univ. (1976). · Zbl 0425.54014
[99] M. Wage, ”On the dimension of product spaces,” Proc. Natl. Acad. Sci. USA, 75, No. 10, 4671–4672 (1978). · Zbl 0387.54019 · doi:10.1073/pnas.75.10.4671
[100] Y. Yajima, ”Topological games and products. I,” Fund. Math., 113, No. 2, 141–153 (1981). · Zbl 0463.54017
[101] Y. Yajima, ”Topological games and products. II,” Fund. Math., 117, No. 1, 47–60 (1983). · Zbl 0519.54002
[102] Y. Yajima, ”On the dimension of limits of inverse systems,” Proc. Amer. Math. Soc., 91, No. 3, 461–466 (1984). · Zbl 0516.54009 · doi:10.1090/S0002-9939-1984-0744649-6
[103] Y. Yajima, ”On {\(\Sigma\)}-products of {\(\Sigma\)}-spaces,” Fund. Math., 123, No. 1, 49–57 (1984).
[104] A. V. Zarelua, ”On the Hurewicz theorem,” Mat. Sb., 60, No. 1, 17–28 (1963). · Zbl 0124.38003
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