ℝ𝑶𝑼𝑮𝑯 𝓸𝑯(𝟏,𝟐)∗ GENERALIZED PRE-HOMEOMORPHISMIN ROUGH BITOPOLOGICAL SPACE

The paper aims to introduce gβ^-closed sets in bitopological spaces. The relationships of (1, 2)-gβ^-closed sets with other kinds of sets are given. The concept union and intersection of (1,2)- gβ^-closed sets are discussed.Also we introduce (1,2)-gβ^-open sets in this paper. The union and intersection on (1,2)-gβ^-open sets are discussed. Also we give some characterizations of (1, 2)-gβ^-closed sets and (1, 2)-gβ^-open sets in bitopological spaces.

In this paper, nano (Ji, Jj )Z open sets, nano (Ji, Jj )Z interior and nano (Ji, Jj )Z closure are introduced in nano (Ji, Jj ) bitopological spaces. Moreover, nano (Ji, Jj )Z-σk continuous functions in nano (Ji, Jj ) bitopological spaces are introduced and some of their characteristics are derived.

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.

In this paper, we introduce the notion of an \((i,j)\)-\(\mathcal{N}\)-\(\beta\)-open set which is a generalization of an \((i,j)\)-\(\beta\)-open set in a bitopological space. Also, we investigate some of its properties and characterizations. Besides, we prove that a pairwise \((i, j)\)-\(\mathcal{N}\)-\(\beta\)-open cover that has a finite (countable) subcover is equivalent to a pairwise \(\beta\)-compact (\(\beta\)-Lindel\"{ö}f) space. Finally, we introduce an \((i, j)\)-\(\mathcal{N}\)-\(\beta\)-continuous function and an \((i, j)\)-\(\mathcal{N}\)-\(\beta\)-irresolute function and obtain some of their properties.

Our aim of this paper is to continue the study of operations on generalized semi open sets in bitopological spaces which was initiated in 2024 by Khedr et al. [12]. New types of pairwise continuous and pairwise irresolute mappings between two bitopological spaces by the concept of operations on generalized semi open sets are introduced with some of their properties and the relation between them are studied. Firstly, we introduce two concepts of pairwise generalized semi operation continuous mappings between two bitopological spaces with some of their properties and study the relation between them. Secondly, we introduce two concepts of pairwise generalized semi operation irresolute mappings between two bitopological spaces with some of their properties and study the relation between them. Finally, we introduce the notions of pairwise generalized semi operation closed mappings and pairwise generalized semi operation open mappings between two bitopological spaces and study some results and theorems.

In many papers, new classes of sets had been studied in topological space, then the notion of continuity between any two topological spaces (a function from X to Y is continuous if the inverse image of each open set of Y is open in X) is studied via this new classes of sets. Here the authors also introduce new classes of sets called pj-b-preopen, pj-b-B set, pj-b-t set, pj-b-semi-open and pj-sb-generalized closed set in bitopological space [1] which is a set with two topologies defined on it, then they study the notion of continuity via this set and introduce some of the theories which are studying the decomposition of continuity via this set in bitopological space.

A NOTE ON COMPUTATION OF NUMBER OF FUZZY BITOPOLOGICAL SPACE

In this paper, we apply the notion of $e$-$\I$-open sets \cite{Wadei6} in ideal topological spaces to present and study new classes of functions called contra $e$-$\I$-continuous functions, almost-$e$-$\I$-continuous, almost contra-$e$-$\I$-continuous, and almost weakly-$e$-$\I$-continuous along with their several properties, characterizations and mutual relationships. Relationships between their new classes and other classes of functions are established and some characterizations of their new classes of functions are studied. Further, we introduce new types of graphs, called $e$-$\I$-closed, contra-$e$-$\I$-closed, and strongly contra-$e$-$\I$-closed graphs via $e$-$\I$-open sets. Several characterizations and properties of such notions are investigated.

The main aim of this paper is to study new class of continuous function is called – continuous function. for this aim, the nation of - open and pre – open sets and - compact space are introduced. and we shall study the relationship between – continuous and pc– continuous functions.

This paper introduces a new class of functions between bitopological spaces, namely contra-$(\tau_1,\tau_2)$-continuous functions. Furthermore, several characterizations and some properties concerning contra-$(\tau_1,\tau_2)$-continuous functions are investigated.

Generalized closed sets play an important role in the study of topological spaces. In the literature we have seen the relation between closed set and generalized closed set is linear in nature, both in crisp and fuzzy environment. In particular, every closed set is a generalized closed set. In this present treatise, a new type of generalized closed set via $$(i,j)^{*}$$ -fuzzy $$\gamma $$ -open set, namely $$(i,j)^{*}$$ - $$\gamma gf\gamma $$ -closed set have been introduced in a fuzzy bitopological space and interrelationships of it with the existing ones have been made. It has been proved that this newly defined generalized fuzzy closed set is totally independent with $$(i,j)^{*}$$ -fuzzy closed set. Various characterizations of this set have been established and some interesting results are cited. As applications of this set various types of continuities are studied in the same environment. More applications are proposed by introducing a new kind of closure operator namely $$(i,j)^{*}$$ - $$\gamma $$ - $$cl_{\gamma }^{*}$$ -operator and important results have been reported based on this concept.

<abstract><p>In this paper, we first introduced the concept of $ r $-fuzzy soft $ \beta $-closed sets in fuzzy soft topological spaces based on the sense of Šostak and investigated some properties of them. Also, we defined the closure and interior operators with respect to the classes of $ r $-fuzzy soft $ \beta $-closed and $ r $-fuzzy soft $ \beta $-open sets and studied some of their properties. Moreover, the concept of $ r $-fuzzy soft $ \beta $-connected sets was introduced and characterized with the help of fuzzy soft $ \beta $-closure operators. Thereafter, some properties of a fuzzy soft $ \beta $-continuity were studied. Also, we introduced and studied the concepts of fuzzy soft almost (weakly) $ \beta $-continuous functions, which are weaker forms of a fuzzy soft $ \beta $-continuity. The relationships between these classes of functions were specified with the help of some illustrative examples. Finally, we explored new types of fuzzy soft functions called fuzzy soft $ \beta $-irresolute (strongly $ \beta $-irresolute, $ \beta $-irresolute open, $ \beta $-irresolute closed, and $ \beta $-irresolute homeomorphism) functions and discussed some properties of them. Also, we showed that fuzzy soft strongly $ \beta $-irresolute $ \Rightarrow $ fuzzy soft $ \beta $-irresolute $ \Rightarrow $ fuzzy soft $ \beta $-continuity, but the converse may not be true.</p></abstract>

In this paper we investigate the existence and uniqueness of weak solutions of the nonautonomous Hamilton–Jacobi–Bellman equation on the domain $$(0,\infty ) \times \Omega $$ . The Hamiltonian is assumed to be merely measurable in time variable and the open set $$\Omega $$ may be unbounded with nonsmooth boundary. The set $$\overline{\Omega }$$ is called here a state constraint. When state constraints arise, then classical analysis of Hamilton–Jacobi–Bellman equation lacks appropriate notion of solution because continuous solutions could not exist. In this work we propose a notion of weak solution for which, under a suitable controllability assumption, existence and uniqueness theorems are valid in the class of lower semicontinuous functions vanishing at infinity.

This paper attempts to determine the “largest” sets of local univalence for a given class and the “largest” open sets in which all functions belonging to a given class are univalent. We establish general properties of sets of univalence for analytic functions with typical normalization. Moreover, we determine some examples of the sets of univalence for some particular subclasses of analytic functions.

Let X be a compact Hausdorff space, C(X) the class of all continuous functions f: X→R, considered as an ordered vector space over R with respect to its canonical order; furthermore let F be an order complete ordered vector space over R. A linear operator A: C(X)→F is called bounded, if it transforms each bounded set of C(X) into a bounded set of F. The purpose of this paper is to represent such a bounded linear operator A: C(X)→F as an integral Af=∫ dμ with respect to some content μ, defined on the algebraa(X) which is generated by the open sets U⊂X. This representation is unique, if μ is required to be regular (p. 11).