A Study on Dimensions of Continuous Mappings

Dimitrios Georgiou; Yasunao Hattori; Athanasios Megaritis; Fotini Sereti

doi:10.2478/tmmp-2025-0013

Abstract

In Dimension Theory, there are “mapping theorems” that establish relationships between the dimensions of a domain and the range of a continuous mapping. Most of the theorems deal with mappings that satisfy additional conditions, such as being closed mappings. In the “environment” of such studies, the dimensions of continuous mappings have also been studied. In this paper, we introduce and investigate a new notion of dimension for continuous mappings between topological spaces, which is closer to the classical definition of the Lebesgue covering dimension of a space. We discuss various results concerning this dimension. Moreover, we present open questions and proposals of new dimensions of continuous mappings for further studies. They are based on known dimensions of continuous mappings between topological spaces.

References

AARTS, J. M.—NISHIURA, T.: Dimension and extensions, North-Holland Mathematical Library, Vol. 48. North-Holland Publishing Co., Amsterdam, 1993.
Search in Google Scholar Back to article
AMOR, A. B.—LAZAAR, S.—RICHMOND, T.—SABRI, H.: k-primal spaces, Topology Appl. 309 (2022), Paper no. 107907, pp. 14.
Search in Google Scholar Back to article
ARHANGEL’SKIĬ, A.: On open and almost-open mappings of topological spaces,Dokl. Akad. Nauk SSSR 147 (1962), 999–1002. (In Russian)
Search in Google Scholar Back to article
BOGATYI, S.—FEDORCHUK, V.—VAN MILL, J.: On mappings of compact spaces into Cartesian spaces, Topol. Appl. 107 (2000), 13–24.
Search in Google Scholar Back to article
CHARALAMBOUS, M. G.: Dimension Theory. A Selection of Theorems and Counterexamples, Atlantis Studies in Mathematics, vol. 7. Springer, Cham, 2019.
Search in Google Scholar Back to article
CHATYRKO, V. A.—PASYNKOV, B. A.: On sum and product theorems for dimension Dind, Houston J. Math. 28 (2002), no. 1, 119–131.
Search in Google Scholar Back to article
CHATYRKO, V. A.: Some questions about dimension Dind, Questions Answers Gen. Topology 40 (2022), no. 1, 1–9.
Search in Google Scholar Back to article
DRANISHNIKOV, A. N.: The Eilenberg-Borsuk theorem for maps into arbitrary complexes, Math. Sb. 185 (1994), no. 4, 81–90.
Search in Google Scholar Back to article
DRANISHNIKOV, A.—REPOV, D.—ŠČEPIN, E.: On intersections of compacta of complementary dimensions in Euclidean space, Topol. Appl. 38 (1991), 237–253.
Search in Google Scholar Back to article
DUBE, T.—ILIADIS, S.—JAN VAN MILL,—NAIDOO, I.: Universal frames, Topology Appl. 160 (2013), 2454–2464.
Search in Google Scholar Back to article
EGOROV, V.—PODSTAVKIN, JU.: A definition of dimension Dokl. Akad. Nauk SSSR 178 (1968), 774–777. (In Russian)
Search in Google Scholar Back to article
EILENBERG, S.: Remarque sur un theorem de W. Hurewicz, Fund. Math. 24 (1935), 156–159.
Search in Google Scholar Back to article
ENGELKING, R.: General Topology, Second edition, Sigma Series in Pure Mathematics Vol. 6. Heldermann Verlag, Berlin, 1989.
Search in Google Scholar Back to article
ENGELKING, R.: Theory of Dimensions, Finite and Infinite, Sigma Series in Pure Mathematics Vol. 10. Heldermann Verlag, Lemgo, 1995.
Search in Google Scholar Back to article
GEORGIOU, D. N.—HATTORI, Y.—MEGARITIS, A. C.—SERETI, F.: The dimension Dind of Finite Topological T₀-spaces, Math. Slov. 72 (2022), no. 3, 813–829.
Search in Google Scholar Back to article
GEORGIOU, D. N.—MEGARITIS, A. C.—SERETI, F.: A topological dimension greater than or equal to the classical covering dimension, Houston J. Math. 43 (2017), no. 1, 283–298.
Search in Google Scholar Back to article
DE GROOT, J.: Topologische Studiën, Compactificatie, Voortzetting van Afbeeldingen en Samenhang. [Topological Studies. Compactification, Extension of Mappings and Connectivity.] Thesis. University of Groningen, Groningen, 1942. 102 pp. (In Dutch)
Search in Google Scholar Back to article
HUREWICZ, W.:Über Abbildungen von endlichdimensionalen R¨aumen auf Teilmengen Cartesischer R¨aume, Sgb. Preuss. Akad. 34 (1933), 754–768.
Search in Google Scholar Back to article
HUREWICZ, W.—WALLMAN, H.: Dimension Theory, Princeton Mathematical Series Vol. 4. Princeton University Press, Princeton, NJ, 1941.
Search in Google Scholar Back to article
ILIADIS, S.: Universal Spaces and Mappings, North-Holland Mathematics Studies Vol. 198. Elsevier Science B. V., Amsterdam, 2005.
Search in Google Scholar Back to article
ILIADIS, S.: Mappings and universality, Topology Appl. 137 (2004), no. 1–3, 175–186.
Search in Google Scholar Back to article
KEESLING, J.: Closed mappings which lower dimension, Colloq. Math. 20 (1969), 237–241.
Search in Google Scholar Back to article
KRZEMPEK, J.: Another approach to dimension of maps, Bull. Polish Acad. Sci. Math. 50 (2002), no. 2, 141–154.
Search in Google Scholar Back to article
KRZEMPEK, J.: Finite-to-one maps and dimension, Fund. Math. 182 (2004), 95–106.
Search in Google Scholar Back to article
KULPA, W.: A note on the dimension Dind, Colloq. Math. 24 (1971/72), 181–183.
Search in Google Scholar Back to article
LELEK, A.: Dimension and mappings of spaces with finite deficiency, Colloq. Math. 12 (1964), 221–227.
Search in Google Scholar Back to article
LEVIN, M.: Bing maps and finite-dimensional maps, Fund. Math. 151 (1996), 47–52.
Search in Google Scholar Back to article
LEVIN, M.: On extensional dimension of maps, Topol. Appl. 103 (2000), 33–35.
Search in Google Scholar Back to article
LEVIN, M.—LEWIS, W.: Some mapping theorems for extensional dimension,Israel J. Math. 133 (2003), 61–76.
Search in Google Scholar Back to article
MENIX, J.—RICHMOND, T.: The lattice of functional Alexandroff topologies,Order 38 (2021), no. 1, 1–11.
Search in Google Scholar Back to article
MORITA, K.: On closed mappings and dimension, Proceedings of the Japan Academy 32 (1956), 161–165.
Search in Google Scholar Back to article
NAGAMI, K.: Some theorems in dimension theory for non-separable spaces, Journal of the Mathematical Society of Japan 7 (1957), 80–92.
Search in Google Scholar Back to article
NAGAMI, K.: Dimension Theory, Pure and Applied Mathematics, Vol. 37, Academic Press, New York-London, 1970.
Search in Google Scholar Back to article
NAGATA, J.: Modern Dimension Theory. (Revised edition). Sigma Series in Pure Mathematics, Vol 2. Heldermann Verlag, Berlin, 1983.
Search in Google Scholar Back to article
NISHIURA, T.: On the dimension of semi-compact spaces and their quasicomponents, Colloq. Math. 12 (1964), 7–10.
Search in Google Scholar Back to article
PASYNKOV, B. A.: On universal compact spaces of given dimension, Fund. Math. 60 (1967), 285–308.
Search in Google Scholar Back to article
PASYNKOV, B. A.: On the dimension and geometry of mappings, Dokl. Akad. Nauk SSSR 221 (1975), 543–546. (In Russian)
Search in Google Scholar Back to article
PASYNKOV, B. A.: On the dimension of rectangular products,SovietMath. Dokl. 16 (1975), 344–347.
Search in Google Scholar Back to article
PASYNKOV, B. A.: Extension to mappings of certain notions and assertions concerning spaces, Mappings and Functors Vol. 143, (1984) 72–102, Izdat. MGU, Moscow, 1984 (In Russian)
Search in Google Scholar Back to article
PASYNKOV, B. A.: On geometry of continuous maps of finite-dimensional compact metric spaces, Trudy Steklov Mat. Inst. 212 (1996), 147–172. (In Russian)
Search in Google Scholar Back to article
PASYNKOV, B. A.: On the geometry of continuous mappings of finite-dimensional metrizable compacta, Tr. Mat. Inst. Steklova 212 (1996), Mappings and Dimension, 147–172. (In Russian)
Search in Google Scholar Back to article
PASYNKOV, B. A.: On geometry of continuous maps of countable functional weight, Fundam. Prikl. Mat. 4 (1998), 155–164. (In Russian)
Search in Google Scholar Back to article
PEARS, A. R.: Dimension Theory of General Spaces, Cambridge University Press, Cambridge, England-New York-Melbourne, 1975.
Search in Google Scholar Back to article
REICHAW, M.: On a theorem of W. Hurewicz on mappings which lower dimension, Colloq. Math. 26 (1972), 323–329.
Search in Google Scholar Back to article
STERNFELD, Y.: On finite-dimensional maps and other maps with “small” fibers, Fund. Math. 147 (1995), 127–133.
Search in Google Scholar Back to article
TUNCALI, H. M.—VALOV, V.: On dimensionally restricted maps, Fund. Math. 175 (2002), 35–52.
Search in Google Scholar Back to article
TUNCALI, H. M.—VALOV, V.: On finite-dimensional maps II, Topol. Appl. 132 (2003), 81–87.
Search in Google Scholar Back to article
TUNCALI, H. M.—VALOV, V.: On finite-dimensional maps, Tsukura J. Math. 28 (2004), no. 1, 155–167.
Search in Google Scholar Back to article
TUNCALI, H. M.—VALOV, V.: On Finite-to-One Maps, Canad. Math. Bull. Vol. 48 (2005), no. 4, 614–621.
Search in Google Scholar Back to article
TUNCALI, H. M.—VALOV, V.: On regularly branched maps, Topol. Appl. 150 (2005), 213–221.
Search in Google Scholar Back to article
TORUŃCZYK, H.: Finite to one restrictions of continuous functions, Fund. Math. 75 (1985), 237–249.
Search in Google Scholar Back to article
TURYGIN, Y. A.: Approximation of k-dimensional maps, Topol. Appl. 139 (2004), 227–235.
Search in Google Scholar Back to article
VALOV, V.: Maps with dimensionally restricted fibers, Colloq. Math. 123 (2011), 239–248.
Search in Google Scholar Back to article
WILLIAMS, R.: The effect of maps upon the dimension of subsets of the domain space, Proc. Amer. Math. Soc. 8 (1957), 580–583.
Search in Google Scholar Back to article

A Study on Dimensions of Continuous Mappings

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