Some Properties of Maximal Open Sets

Some fundamental properties of maximal open sets are obtained, such as decomposition theorem for a maximal open set. Basic properties of intersections of maximal open sets are established, such as the law of radical closure. 1. Introduction. A proper nonempty open subset U of a topological space X is said to be a maximal open set if any open set which contains U is X or U. In [2], we study minimal open sets. Although the definition of the maximal open set is obtained by " dualizing " the definition of the minimal open set, the properties of them are quite different, as we see in this paper, especially the results in the last two sections. The purpose of this paper is to prove some fundamental properties of maximal open sets and establish a part of the foundation of the theory of maximal open sets in topological spaces. In Section 2, we prove some basic results which are necessary for the subsequent arguments. We obtain a relation among maximal open sets in Theorem 2.5. At the end of this section, we show that for any proper nonempty cofinite open subset V , there exists, at least, one maximal open set U which contains V (Theorem 2.7). In Section 3, we study some relations among closure, interior, and maximal open sets. As an application, we prove a result about a preopen set (Theorem 3.11). Let ᐁ = {U λ | λ ∈ Λ} be a set of some maximal open sets U λ. Then, we refer to the intersection ∩ᐁ = ∩ λ∈Λ U λ as the radical of ᐁ. In the last two sections, we study various properties of radicals. In Section 4, we prove fundamental properties of radicals of maximal open sets. We establish a very useful decomposition theorem for a maximal open set in Theorem 4.7. Theorem 4.7 will be applied to prove Theorem 4.8. Theorem 4.9 gives a sufficient condition for the set of all maximal open sets. In the rest of this section, we study the case when radicals are closed sets. In Section 5, we consider the closure of the radicals of maximal open sets. We establish " The law of radical closure " in Theorem 5.4.