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Topologies on Zn that Are Not Homeomorphic to the n-Dimensional Khalimsky Topological Space

1
Department of Mathematics Education, Institute of Pure and Applied Mathematics Jeonbuk National University, Jeonju-City 54896, Jeonbuk, Korea
2
College of Vestsjaelland South Herrestraede 114200 Slagelse, Denmark
3
Mathematics, School of Liberal, Arts Education, University of Seoul, Seoul 02504, Korea
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(11), 1072; https://doi.org/10.3390/math7111072
Received: 6 October 2019 / Revised: 1 November 2019 / Accepted: 4 November 2019 / Published: 7 November 2019
The present paper deals with two types of topologies on the set of integers, Z : a quasi-discrete topology and a topology satisfying the T 1 2 -separation axiom. Furthermore, for each n N , we develop countably many topologies on Z n which are not homeomorphic to the typical n-dimensional Khalimsky topological space. Based on these different types of new topological structures on Z n , many new mathematical approaches can be done in the fields of pure and applied sciences, such as fixed point theory, rough set theory, and so on.
Keywords: khalimsky topology; quasi-discrete (clopen or pseudo-discrete); T12-separation axiom; alexandroff topology; digital topology khalimsky topology; quasi-discrete (clopen or pseudo-discrete); T12-separation axiom; alexandroff topology; digital topology
MDPI and ACS Style

Han, S.-E.; Jafari, S.; Kang, J.M. Topologies on Zn that Are Not Homeomorphic to the n-Dimensional Khalimsky Topological Space. Mathematics 2019, 7, 1072.

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