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.
Highlights
This present paper has been written based on the first author’s 2016 paper [1], and the present paper has been completed with many enhancements and extensions of the previous paper [1].In particular, the sufficient conditions of the previous paper have been changed to necessary and sufficient conditions in this paper, which greatly improved the completeness of this paper.of the present paper, we introduce some necessary and sufficient conditions that the intersection of an open set and a closed set of a topological space becomes either an open set or a closed set, even though it seems to be a typically classical subject
The sufficient conditions of the previous paper have been changed to necessary and sufficient conditions in this paper, which greatly improved the completeness of this paper
Roughly speaking, we prove that the intersection of a connected open set and a closed set is open if and only if the closed set includes the open set
Summary
The sufficient conditions of the previous paper have been changed to necessary and sufficient conditions in this paper, which greatly improved the completeness of this paper. We present some necessary and sufficient conditions that the union of a closed set and an open set becomes either a closed set or an open set. The inclusion (1) or (2) holds true for ‘strict inclusion’ when U and V are some particular subsets of a topological space X.
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