| Summary: | 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 <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>U</mi> <mo>∘</mo> </msup> <mo>∪</mo> <msup> <mi>V</mi> <mo>∘</mo> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mi>U</mi> <mo>∪</mo> <mi>V</mi> <mo>)</mo> </mrow> <mo>∘</mo> </msup> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mover> <mi>U</mi> <mo>¯</mo> </mover> <mo>∩</mo> <mover> <mi>V</mi> <mo>¯</mo> </mover> <mo>=</mo> <mover> <mrow> <mi>U</mi> <mo>∩</mo> <mi>V</mi> </mrow> <mo>¯</mo> </mover> </mrow> </semantics> </math> </inline-formula>. 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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