An algorithm of polynomial order for computing the covering dimension of a finite space

“…The Dimension Theory is a developing branch of Topology, which attracts the interest of many researches (see for example [1,2,[4][5][6][7][8][9][10][11][12][13]18]). Especially, the covering dimension, dim, the small inductive dimension, ind, and the large inductive dimension, Ind, are three main topological dimensions, which have been studied extensively, and many results in various classes of topological spaces have been proved.…”

Section: Introductionmentioning

confidence: 99%

Small and large inductive dimension for ideal topological spaces

Sereti

2021

Appl. Gen. Topol.

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<p>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.</p>

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“…As main result it is shown that the dimension ind(X, T ) of an Alexandroff T 0 -space (X, T ) equals the height of the specialisation order of T . This work is continued in [2,10,11,12,13,14,15], thereby generalising it to Alexandroff spaces which are not T 0 -spaces and to the other two important notions of a dimension in topology. The latter are the large inductive dimension Ind and the covering dimension dim.…”

Section: Introductionmentioning

confidence: 99%

Algorithmic Counting of Zero-Dimensional Finite Topological Spaces With Respect to the Covering Dimension

Berghammer¹,

Börm²,

Winter³

2020

Preprint

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Taking the covering dimension dim as notion for the dimension of a topological space, we first specify the number zdim T 0 (n) of zero-dimensional T 0 -spaces on {1, . . . , n} and the number zdim(n) of zero-dimensional arbitrary topological spaces on {1, . . . , n} by means of two mappings po and P that yield the number po(n) of partial orders on {1, . . . , n} and the set P(n) of partitions of {1, . . . , n}, respectively. Algorithms for both mappings exist. Assuming one for po to be at hand, we use our specification of zdim T 0 (n) and modify one for P in such a way that it computes zdim T 0 (n) instead of P(n). The specification of zdim(n) then allows to compute this number from zdim T 0 (1) to zdim T 0 (n) and the Stirling numbers of the second kind S (n, 1) to S (n, n). The resulting algorithms have been implemented in C and we also present results of practical experiments with them. To considerably reduce the running times for computing zdim T 0 (n), we also describe a backtracking approach and its parallel implementation in C using the OpenMP library.

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An algorithm of polynomial order for computing the covering dimension of a finite space

Cited by 12 publications

References 11 publications

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Small and large inductive dimension for ideal topological spaces

Algorithmic Counting of Zero-Dimensional Finite Topological Spaces With Respect to the Covering Dimension

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