Some Properties of Interior and Closure in General Topology

Soon-Mo Jung and Doyun Nam
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Soon-Mo Jung: Mathematics Section, College of Science and Technology, Hongik University, Sejong 30016, Korea
Doyun Nam: Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea

Mathematics, 2019, vol. 7, issue 7, 1-10

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 ? ∪ V ? = ( U ∪ V ) ? and U ¯ ∩ V ¯ = U ∩ V ¯ . 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.

Keywords: open set; closed set; duality; union; intersection; topological space (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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