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

Small Inductive Dimension of Topological Spaces. Part II

In this paper we present basic properties of n-dimensional topological spaces according to the book [10]. In the article the formalization of Section 1.5 is completed.Institute of Computer Science, University of Białystok, PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481-485, 1991.Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Ryszard Engelking. Teoria wymiaru. PWN, 1981.Robert Milewski. Bases of continuous lattices. Formalized Mathematics, 7(2):285-294, 1998.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Karol Pąk. Small inductive dimension of topological spaces. Formalized Mathematics, 17(3):207-212, 2009, doi: 10.2478/v10037-009-0025-7.Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535-545, 1991.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231-237, 1990

On dimension of topological space

U ovom radu definiramo tri dimenzije topološkog prostora, malu induktivnu dimenziju, veliku induktivnu dimenziju i dimenziju pokrivanja, te dokazujemo neke važne teoreme o tim dimenzijama. Posebno, dokazujemo da se u klasi seprabilnih metrizabilnih prostora podudaraju sve tri dimenzije, te da se u klasi metrizabilnih prostora podudaraju velika induktivna dimenzija i dimenzija pokrivanja. Naposljetku, dokazujemo da je dimenzija euklidskog prostora Rn jednaka n, bilo da se radi o maloj induktivnoj dimenziji, velikoj induktivnoj dimenziji, ili dimenziji pokrivanja.In this thesis we define three different dimensions of a topological space, small inductive dimension, large inductive dimension and covering dimension, and we prove some important theorems about these dimensions. In particular, we prove that the three dimensions coincide in the class of all separable metrizable spaces, while the large inductive dimension and the covering dimension coincide in the class of all metrizable spaces. Finally, we prove that the dimension of the Euclidean space Rn is equal to n, whether it is the small inductive dimension, the large inductive dimension, or the covering dimension

On dimension of topological space

U ovom radu definiramo tri dimenzije topološkog prostora, malu induktivnu dimenziju, veliku induktivnu dimenziju i dimenziju pokrivanja, te dokazujemo neke važne teoreme o tim dimenzijama. Posebno, dokazujemo da se u klasi seprabilnih metrizabilnih prostora podudaraju sve tri dimenzije, te da se u klasi metrizabilnih prostora podudaraju velika induktivna dimenzija i dimenzija pokrivanja. Naposljetku, dokazujemo da je dimenzija euklidskog prostora Rn jednaka n, bilo da se radi o maloj induktivnoj dimenziji, velikoj induktivnoj dimenziji, ili dimenziji pokrivanja.In this thesis we define three different dimensions of a topological space, small inductive dimension, large inductive dimension and covering dimension, and we prove some important theorems about these dimensions. In particular, we prove that the three dimensions coincide in the class of all separable metrizable spaces, while the large inductive dimension and the covering dimension coincide in the class of all metrizable spaces. Finally, we prove that the dimension of the Euclidean space Rn is equal to n, whether it is the small inductive dimension, the large inductive dimension, or the covering dimension

Hurewicz fibrations, almost submetries and critical points of smooth maps

We prove that the existence of a Hurewicz fibration between certain spaces with the homotopy type of a CW-complex implies some topological restrictions on their universal coverings. This result is used to deduce differentiable and metric properties of maps between compact Riemannian manifolds under curvature restrictions

The topological dimension of type I C*-algebras

While there is only one natural dimension concept for separable, metric spaces, the theory of dimension in noncommutative topology ramifies into different important concepts. To accommodate this, we introduce the abstract notion of a noncommutative dimension theory by proposing a natural set of axioms. These axioms are inspired by properties of commutative dimension theory, and they are for instance satisfied by the real and stable rank, the decomposition rank and the nuclear dimension. We add another theory to this list by showing that the topological dimension, as introduced by Brown and Pedersen, is a noncommutative dimension theory of type I C*-algebras. We also give estimates of the real and stable rank of a type I C*-algebra in terms of its topological dimension.Comment: 20 pages; minor correction