A study of the quasi covering dimension of Alexandroff countable spaces using matrices

Abstract

The notion of Alexandroff space was, firstly, appeared in [1]. It is known that such spaces have all the properties of finite topological spaces and they play an essential role in digital topology. Also, different types of the so called covering dimension, in the class of all Alexandroff countable spaces have been studied (see, [2]). However, the study of the notion of dimension in such classes of spaces is, still, an open field for further research. Thus, inspired by [3], where a new topological dimension, called quasi covering dimension, was inserted, in this paper, we study this new dimension in the class of all Alexandroff countable topological spaces using the matrix algebra. Especially, we characterize the open and dense subsets of an arbitrary Alexandroff countable space X using matrices and, based on these characterizations, we provide a computational procedure for the determination of the quasi covering dimension of X.

References

[1] P. Alexandroff, Diskrete Raume, Mat. Sb. 2 (1937) 501-518.

[2] D. N. Georgiou, S.-E. Han, A. C. Megaritis, Dimensions of the type dim and Alexandroff spaces, Journal of Egyptian Mathematical Society 21 (2013) 311-317.

[3] D. N. Georgiou, A. C. Megaritis, F. Sereti, A topological dimension greater than or equal to the classical covering dimension, Houston Journal of Mathematics 43(1) (2017) 283-298.