Abstract
Some properties of minimal closed sets and maximal closed sets are obtained, which are dual concepts of maximal open sets and minimal open sets, respectively. Common properties of minimal closed sets and minimal open sets are clarified; similarly, common properties of maximal closed sets and maximal open sets are obtained. Moreover, interrelations of these four concepts are studied.
Highlights
Some properties of minimal open sets and maximal open sets are studied in [1, 2]
We define dual concepts of them, namely, maximal closed set and minimal closed set. These four types of subsets appear in finite spaces, for example
Minimal open sets and maximal closed sets appear in locally finite spaces such as the digital line
Summary
We define dual concepts of them, namely, maximal closed set and minimal closed set These four types of subsets appear in finite spaces, for example. We have to consider only “proper nonempty open set U of X,” as in the following definitions: a proper nonempty open subset U of X is said to be a minimal open set if any open set which is contained in U is ∅ or U. A proper nonempty open subset U of X is said to be a maximal open set if any open set which contains U is X or U. A proper nonempty closed subset F of X is said to be a maximal closed set if any closed set which contains F is X or F.
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