Some Properties of Interior and Closure in General Topology
- Authors
- Jung, Soon-Mo; Nam, Doyun
- Issue Date
- Jul-2019
- Publisher
- MDPI
- Keywords
- open set; closed set; duality; union; intersection; topological space
- Citation
- MATHEMATICS, v.7, no.7, pp.1 - 10
- Journal Title
- MATHEMATICS
- Volume
- 7
- Number
- 7
- Start Page
- 1
- End Page
- 10
- URI
- https://scholarworks.bwise.kr/hongik/handle/2020.sw.hongik/1377
- DOI
- 10.3390/math7070624
- ISSN
- 2227-7390
- Abstract
- We present the necessary and sufficient conditions that the intersection of an open set and a closed set becomes either an open set or a closed set. As their dualities, we further introduce the necessary and sufficient conditions that the union of a closed set and an open set becomes either a closed set or an open set. Moreover, we give some necessary and sufficient conditions for the validity of U circle?V circle=(U?V)circle and U over bar boolean AND V over bar =U boolean AND V over bar . Finally, we introduce a necessary and sufficient condition for an open subset of a closed subspace of a topological space to be open. As its duality, we also give a necessary and sufficient condition for a closed subset of an open subspace to be closed.
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