Small and large inductive dimension for ideal topological spaces

[EN] Undoubtedly, the small inductive dimension, ind, and the large inductive dimension, Ind, for topological spaces have been studied extensively, developing an important field in Topology. Many of their properties have been studied in details (see for example [1,4,5,9,10,18]). However, researches for dimensions in the field of ideal topological spaces are in an initial stage. The covering dimension, dim, is an exception of this fact, since it is a meaning of dimension, which has been studied for such spaces in [17]. In this paper, based on the notions of the small and large inductive dimension, new types of dimensions for ideal topological spaces are studied. They are called *-small and *-large inductive dimension, ideal small and ideal large inductive dimension. Basic properties of these dimensions are studied and relations between these dimensions are investigated.Sereti, F. (2021). Small and large inductive dimension for ideal topological spaces. Applied General Topology. 22(2):417-434. https://doi.org/10.4995/agt.2021.15231OJS417434222M. G. Charalambous, Dimension Theory, A Selection of Theorems and Counterexample, Springer Nature Switzerland AG, Cham, Switzerland, 2019. https://doi.org/10.1007/978-3-030-22232-1M. Coornaert, Topological Dimension, In: Topological dimension and dynamical systems, Universitext. Springer, Cham, 2015. https://doi.org/10.1007/978-3-319-19794-4J. Dontchev, M. Maximilian Ganster and D. Rose, Ideal resolvability, Topology Appl. 93, no. 1 (1999), 1-16. https://doi.org/10.1016/S0166-8641(97)00257-5R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.R. Engelking, Theory of Dimensions, Finite and Infinite, Heldermann Verlag, Berlin, 1995.D. N. Georgiou, S. E. Han and A. C. Megaritis, Dimensions of the type dim and Alexandroff spaces, J. Egypt. Math. Soc. 21 (2013), 311-317. https://doi.org/10.1016/j.joems.2013.02.015D. N. Georgiou and A. C. Megaritis, An algorithm of polynomial order for computing the covering dimension of a finite space, Applied Mathematics and Computation 231 (2014), 276-283. https://doi.org/10.1016/j.amc.2013.12.185D. N. Georgiou and A. C. Megaritis, Covering dimension and finite spaces, Applied Mathematics and Computation 218 (2014), 3122-3130. https://doi.org/10.1016/j.amc.2011.08.040D. N. Georgiou, A. C. Megaritis and S. Moshokoa, A computing procedure for the small inductive dimension of a finite $T_0$ space, Computational and Applied Mathematics 34, no. 1 (2015), 401-415. https://doi.org/10.1007/s40314-014-0125-zD. N. Georgiou, A. C. Megaritis and S. Moshokoa, Small inductive dimension and Alexandroff topological spaces, Topology Appl. 168 (2014), 103-119. https://doi.org/10.1016/j.topol.2014.02.014D. N. Georgiou, A. C. Megaritis and F. Sereti, A study of the quasi covering dimension for finite spaces through matrix theory, Hacettepe Journal of Mathematics and Statistics 46, no. 1 (2017), 111-125.D. N. Georgiou, A. C. Megaritis and F. Sereti, A study of the quasi covering dimension of Alexandroff countable spaces using matrices, Filomat 32, no. 18 (2018), 6327-6337. https://doi.org/10.2298/FIL1818327GD. N. Georgiou, A. C. Megaritis and F. Sereti, A topological dimension greater than or equal to the classical covering dimension, Houston Journal of Mathematics 43, no. 1 (2017), 283-298.T. R. Hamlett, D. Rose and D. Janković, Paracompactness with respect to an ideal, Internat. J. Math. Math. Sci. 20, no. 3 (1997), 433-442. https://doi.org/10.1155/S0161171297000598D. Janković and T. R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly 97, no. 4 (1990), 295-310. https://doi.org/10.1080/00029890.1990.11995593K. Kuratowski, Topologie I, Monografie Matematyczne 3, Warszawa-Lwów, 1933.A. C. Megaritis, Covering dimension and ideal topological spaces, Quaestiones Mathematicae, to appear. https://doi.org/10.2989/16073606.2020.1851309A. R. Pears, Dimension theory of general spaces, Cambridge University Press, Cambridge, 1975.P. Samuels, A topology formed from a given topology and ideal, J. London Math. Soc. 10, no. 4 (1975), 409-416. https://doi.org/10.1112/jlms/s2-10.4.40

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