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Some Properties of Interior and Closure in General Topology

Author

Listed:
  • Soon-Mo Jung

    (Mathematics Section, College of Science and Technology, Hongik University, Sejong 30016, Korea
    These authors contributed equally to this work.)

  • Doyun Nam

    (Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Soon-Mo Jung & Doyun Nam, 2019. "Some Properties of Interior and Closure in General Topology," Mathematics, MDPI, vol. 7(7), pages 1-10, July.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:7:p:624-:d:248210
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