Maximal Open Set Research Articles

Some results on maximal open sets

In this paper, the notion of maximal m-open set is introduced and its properties are investigated. Some results about existence of maximal m-open sets are given. Moreover, the relations between maximal m-open sets in an m-space and maximal open sets in the corresponding generated topology are considered. Our results are supported by examples and counterexamples.

On Pairwise Minimal Open Sets In Bitopological Spaces

In this paper, a new class of sets called pairwise minimal open sets and pairwise maximal open sets in bitopological spaces are introduced and studied. A proper nonempty \(\tau_{i}\)-open subset \(M\) of a bitopological space \(X\) is said to be a pairwise minimal open (resp. pairwise maximal open) set if any \(\tau_{j}\)-open set which is contained in \(M\) (resp. contains \(M\)) is either \(\phi\) ( resp. either \(X\) ) or \(M\) itself.

ON SOME NEW MAXIMAL AND MINIMAL SETS VIA θ-OPEN SETS

Abstract. Nakaoka and Oda ([1] and [2]) introduced the notion of max-imal open sets and minimal closed sets in topological spaces. In thispaper, we introduce new classes of sets called maximal θ -open sets, min-imal θ -closed sets, θ -semi maximal open and θ -semi minimal closed andinvestigate some of their fundamental properties. 1. Introduction and preliminariesGeneralized open sets play a very important role in General Topology andthey are now the research topics of many topologists worldwide. Indeed asignificant theme in General Topology and Real Analysis concerns the variouslymodified forms of continuity, separation axioms etc by utilizing generalizedopen sets. One of the most well-known notions and also an inspiration sourceis the notion of θ -open sets introduced by N. V. Veli˘cko [3] in 1968. Since thecollection of θ -open sets in a topological space ( X,τ ) forms a topology τ θ on X then the union of two θ -open sets is of course θ -open. Moreover τ = τ θ if andonly if ( X,τ ) is regular.F. Nakaoka and N. Oda in [1] and [2] introduced the notion of maximal opensets and minimal closed sets. The purpose of the present paper is to introducethe concept of a new class of open sets called maximal

Asymptotic behavior and zero distribution of Carleman orthogonal polynomials

Let L be an analytic Jordan curve and let { p n ( z ) } n = 0 ∞ be the sequence of polynomials that are orthonormal with respect to the area measure over the interior of L . A well-known result of Carleman states that (1) lim n → ∞ p n ( z ) ( n + 1 ) / π [ ϕ ( z ) ] n = ϕ ′ ( z ) locally uniformly on a certain open neighborhood of the closed exterior of L , where ϕ is the canonical conformal map of the exterior of L onto the exterior of the unit circle. In this paper we extend the validity of (1) to a maximal open set, every boundary point of which is an accumulation point of the zeros of the p n ’s. Some consequences on the limiting distribution of the zeros are discussed, and the results are illustrated with two concrete examples and numerical computations.

On Maximal and Minimal Fuzzy Sets in I‐Topological Spaces

The notion of maximal fuzzy open sets is introduced. Some basic properties and relationships regarding this notion and other notions of I‐topology are given. Moreover, some deep results concerning the known minimal fuzzy open sets concept are given.

Open irredundance and maximum degree in graphs

A necessary and sufficient condition for an open irredundant set of vertices of a graph to be maximal is obtained. This result is used to show that the smallest cardinality amongst the maximal open irredundant sets in an n -vertex isolate-free graph with maximum degree Δ is at least n ( 3 Δ − 1 ) / ( 2 Δ 3 − 5 Δ 2 + 8 Δ − 1 ) for Δ ≥ 5 , n / 8 for Δ = 4 and 2 n / 11 for Δ = 3 . The bounds are the best possible.

Envelope of holomorphy for boundary cross sets

Let \(D\subset{\mathbb{C}}^n, G\subset{\mathbb{C}}^m\) be open sets, let A (resp. B) be a subset of the boundary ∂D (resp. ∂G) and let W be the 2-fold boundary cross \(((D\cup A)\times B) \cup (A \times (B \cup G))\). An open subset \(X \subset {{\mathbb{C}}^{n+m}}\) is said to be the “envelope of holomorphy” of W if it is, in some sense, the maximal open set with the following property: Any function locally bounded on W and separately holomorphic on \((A \times G) \cup (D \times B)\) “extends” to a holomorphic function defined on X which admits the boundary values f a.e. on W. In this work we will determine the envelope of holomorphy of some boundary crosses.

Minimal closed sets and maximal closed sets

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.

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.

On Flatness of Generic Projections

Let R be a commutative noetherian ring and let I be an ideal of R[ x 1,…, x n = R [ x]. The morphism ψ R → R [ x]/ I defines a family of algebraic varieties as follows: Let p be a prime ideal of R (or an element of Spec R) and let K( p) be the quotient field of the localization R p of R at p, then we have an algebraic variety in A n K( p) defined by K ( p)[ x]/ I( p) where I( p) = I.K ( p)[ x]. When p varies, these varieties are called fibers of ψ. On the other hand, when ψ is flat, many properties are preserved in the fibers. The main objective of this paper is to characterize flatness of ψ by studying the relationship with the notions of Gröbner and standard bases. When R is principal, we obtain an algorithm to compute the maximal generic open set of flatness of Spec R and then we give some applications related to this situation.

A Note on a Paper of J. D. Stein, Jr.

Among other results, it is proved that if a sequence {t 1l) of regular measures on a Hausdorff space, with values in a normed group, is convergent to zero for all a-compact sets or all open sets, then there exists a maximal open set U such that ft,1 ( U) O-* { f t,) being the associated submeasures. In [5], J. D. Stein, Jr. considers some versions of Phillip's lemma and the Vitali-Hahn-Saks theorem for sequences of regular scalar-valued Borel measures on Hausdorff spaces. The measures he considers are bounded, regular, and finitely additive Borel measures which are easily seen to be countably additive (to prove this, first note that I pi, the variation of such a measure /L, is regular, bounded, and finitely additive [5, Lemma 1]) and so for a sequence {Bi) of Borel sets, B1j0, and > 0, C a sequence of compact sets (Ki), Ki c Bi and vi(Bi K1) some no and so I til(BiWO), an observation which enables one to prove his results easily, in more general forms, and under weaker conditions. Theory of submeasures developed in [1] will be used. Let G be an Abelian Hausdorff topological group, X a Hausdorff topological space, and yt a countably additive, regular, G-valued measure on X (by regularity we mean that for any Borel set B in X and a 0-nbd U in G, there exists a compact set C c B such that tt(K) C U, VK c B C, K Borel). If G is normed [1, p. 270], we have an associated submeasure ft, AK(B) = sup{14(A)I:A c B,A Borel) = sup{ I(C)I: C c B, Ccompact), which is finite [1, p. 279, Corollary 4.1 1], exhaustive, a-subadditive, and order-continuous [1, II]. It is also regular in the sense that given E > 0 and a Borel set B, 3 a compact set K c B such that i(B K) < (proof by contradiction, using exhaustivity). For a collection of measures or subReceived by the editors August 25, 1976. AMS (MOS) subject classifications (1970). Primary 28A25; Secondary 46G10.

A note on a paper of J. D. Stein, Jr.: “Sequence of regular finitely additive set functions” (Trans. Amer. Math. Soc. 192 (1974), 59–66)

Among other results, it is proved that if a sequence { μ n } \{ {\mu _n}\} of regular measures on a Hausdorff space, with values in a normed group, is convergent to zero for all σ \sigma -compact sets or all open sets, then there exists a maximal open set U such that μ ˙ n ( U ) → 0 , { μ ˙ n } {\dot \mu _n}(U) \to 0,\{ {\dot \mu _n}\} being the associated submeasures.