Semi-separation axioms associated with the Alexandroff compactification of the $ MW $-topological plane

Author:

Lee Sik1,Han Sang-Eon2

Affiliation:

1. Department of Mathematics Education, Chonnam National University, Gwangju 61186, Republic of Korea

2. Department of Mathematics Education, Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju, Jeonbuk 54896, Republic of Korea

Abstract

<abstract><p>The present paper aims to investigate some semi-separation axioms relating to the Alexandroff one point compactification (Alexandroff compactification, for short) of the digital plane with the Marcus-Wyse ($ MW $-, for brevity) topology. The Alexandroff compactification of the $ MW $-topological plane is called the infinite $ MW $-topological sphere up to homeomorphism. We first prove that under the $ MW $-topology on $ {\mathbb Z}^2 $ the connectedness of $ X(\subset {\mathbb Z}^2) $ with $ X^\sharp\geq 2 $ implies the semi-openness of $ X $. Besides, for the infinite $ MW $-topological sphere, we introduce a new condition for the hereditary property of the compactness of it. In addition, we investigate some conditions preserving the semi-openness or semi-closedness of a subset of the $ MW $-topological plane in the process of an Alexandroff compactification. Finally, we prove that the infinite $ MW $-topological sphere is a semi-regular space; thus, it is a semi-$ T_3 $-space because it is a semi-$ T_1 $-space. Hence we finally conclude that an Alexandroff compactification of the $ MW $-topological plane preserves the semi-$ T_3 $ separation axiom.</p></abstract>

Publisher

American Institute of Mathematical Sciences (AIMS)

Subject

General Mathematics

Reference36 articles.

1. J. R. Munkres, Topology: A First Course, Pearson College Div, 1974.

2. P. Alexandorff, Uber die Metrisation der im Kleinen kompakten topologischen Räume, Math. Ann., 92 (1924), 294–301.

3. N. Levine, Semi-open sets and semi-continuity in topological spaces, Mathematics, 70 (1963), 36–41. https://doi.org/10.1080/00029890.1963.11990039

4. C. Dorsett, Semi-regular spaces, Soochow J. Math., 8 (1982), 45–53.

5. S. N. Maheshwari, R. Prasad, Some new separation axioms, Ann. Soc. Sci. Bruxelles 89 (1975), 395–402.

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3