Minimal closed sets and maximal closed sets

Some properties of minimal open sets and maximal open sets are studied in [1, 2]. In this paper, 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. More generally, minimal open sets and maximal closed sets appear in locally finite spaces such as the digital line. Minimal closed sets and maximal open sets appear in cofinite topology, for example. We have to study these four concepts to understand the relations among their properties. Some of the results in [1, 2] can be dualized using standard techniques of general topology. But we have to study the “dual results” carefully to understand the duality which we propose in this paper and in [2]. For example, considering the interrelation of these four concepts, we see that some results in [1, 2] can be generalized further. In [1] we called a nonempty open set U of X a minimal open set if any open set which is contained in U is ∅ or U . But to consider duality, 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 . In this paper, we will use the following definitions. A proper nonempty closed subset F of X is said to be aminimal closed set if any closed set which is contained in F is ∅ or F. A proper nonempty closed subset F of X is said to be amaximal closed set if any closed set which contains F is X or F.