Beta- Star- Continuity and Beta- Star- Contra- Continuity

In [3], Dontchev introduced and investigated a new notion of continuity called contra-continuity. Recently, Jafari and Noiri [5] introduced new generalization of contra-continuity called contra continuity . The aim of this paper is to introduce and study the concept of a contra* continuous and almost contra * continuous functions are introduced

In this paper, first we define β*-open sets and β*-interior in topological spaces.J.Antony Rex

A new class of functions, called somewhat -continuous functions, has been defined and studied by making use of -open sets. Characterizations and properties of somewhat -continuous functions are presented.

2015

The aim of this paper is to introduce two new classes of functions, namely slightly *-continuous functions and totally *-continuous functions and study its properties. Mathematics Subject Classification: 54A05 Keywords and phrases: *-open set, *-closed set, slightly *-continuous functions and totally *-continuous functions. * (A) denote the *-closure and the *-interior of A respectively. Definition 2.1: A subset A of a topological space (X,τ) is called a *-open set [6] if there is an open set U in X such that U⊆A⊆ sCl * (U). The collection of all S g *-open sets in (X, τ) is denoted by * O(X,τ). Definition 2.2: A subset A of a topological space (X,τ) is called a *-closed set[6] if X\A is *-open. The collection of all *-closed sets in (X, τ) is denoted by * (X,τ). Theorem 2.3 [6]:Every open set is S g *-open and every closed set is S g *-closed set Definition 2.4: A topological space X, τ is said to be *-space [7] if every *-open set of X is open in X. Definition 2.5: A topological space X, τ is said to be *-locally indiscrete space [8] if every *-open set of X is closed in X. Definition 2.6: A function f: X → Y is said to be contra-*-continuous [8] if the inverse image of every open set in Y is *-closed in X. Definition 2.7: A function f: (X, τ) ⟶ (Y, σ) is called a contra-continuous [2] if f −1 (O) is closed in (X, τ) for every open subset O of (Y, σ). Definition 2.8: A mapping f: X → Y is said to be *-continuous [7] if the inverse image of every open set in Y is *-open in X. Defintion 2.9: A map f: X → Y is said to be *-irresolute[7] if the inverse image of every S g *-open set in Y is *-open in X. Definition 2.10: A mapping f: X → Y is said to be strongly *-continuous [7]if the inverse image of every *open set in Y is open in X. Definition 2.11: A mapping f: X → Y is said to be perfectly *-continuous [7]if the inverse image of every *open set in Y is open and closed in X. Slightly *-continuous functions and Totally *-continuous functions in Topological spaces

Tikrit Journal of Pure Science

In this paper, we study the concepts of contra b-I-continuity and contra b-I-openness in ideal topological spaces, and obtain several characterizations and some properties of two functions. Also, we investigate its relationship with other types of functions.

2008

Abstract. In this paper, we introduce the concept of an operation γ on a family of β-open sets denoted by βO(X) in a topological space (X, τ). Using the operation γ on βO(X), we introduce the concept of β-γ-open sets, and investigate the related topological properties. We also introduce the notion of β-γ-Ti spaces (i = 0, 1/2, 1, 2) and study some topological properties on them. Further, we introduce β-(γ, b)-continuous maps and investigate basic properties. Finally, we investigate a general operation approach to β-closed graphs of mappings.

ISSN: 0973-6571 In 1963 N. Levine [9] and in 1969 N. Biswas [5] have respectively defined and studied the notions called semi open sets and semi closed sets in topology. In the years 1969 and 1970, N. Biswas [4, 5] had defined and studied the notions of semiopen and semiclosed functions respectively. In 1996 J. Dontchev [7] has defined and studied the notion of contra-continuity and in 1997 C.W. Baker [3] has defined the dual notions of contra-continuity, called contra -open functions and contra-closed functions. Then many authors have defined and studied the various concepts of generalized conta – continuity forms in topology. More recently G. Navalagi [13] has studied the notions of contra semipre-open and contra semipre-closed functions. Analogous to Navalagi's results, in this paper we study the generalizations of semi open functions and semi closed functions which are called as contra-semiopen and contra -semiclosed functions respectively. Contra -semiopen functions are cha...

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.

Mathematical Communications, 2011

In this paper, we introduce the concept of an operation γ on a family of β-open sets denoted by βO(X) in a topological space (X, τ). Using the operation γ on βO(X), we introduce the concept of β-γ-open sets, and investigate the related topological properties. We also introduce the notion of β-γ-T i spaces (i = 0, 1/2, 1, 2) and study some topological properties on them. Further, we introduce β-(γ, b)-continuous maps and investigate basic properties. Finally, we investigate a general operation approach to β-closed graphs of mappings.

The Eurasia Proceedings of Science Technology Engineering and Mathematics, 2023

Topology being somehow very recent in nature but has got tremendous applications over almost all other fields. Theoretical or fundamental topology is a bit dry but the application part is what drives crazy once we get used. Topology has applications in various fields of Science and Technology, like applications to Biology, Robotics, GIS, Engineering, Computer Sciences. Topology though being a part of mathematics but it has influenced the whole world with so strong effects and incredible applications. The concept of continuity is fundamental in large parts of contemporary mathematics. In the nineteenth century, precise definitions of continuity were formulated for functions of a real or complex variable, enabling mathematicians to produce rigorous proofs of fundamental theorems of real and complex analysis, such as the Intermediate Value Theorem, Taylor's Theorem, the Fundamental Theorem of Calculus, and Cauchy's Theorem. In the early years of the Twentieth Century, the concept of continuity was generalized so as to be applicable to functions between metric spaces, and subsequently to functions between topological spaces. Topology is an area of mathematics concerned with the properties of space that are preserved under continuous deformations including stretching and bending but not tearing. In 2023, Dr. T. Delcia and M. S, Thillai introduced a new type of closed sets called g**β-closed sets and investigated their basic properties including their relationship with already existing concepts in Topological Spaces. In this paper, we introduce g**β-continuous function, g**β-irresolute function, g**β-open function, g**β-closed function, pre-g**β-open function, and pre-g**β-closed function, and investigate properties and characterizations of these new types of mappings in topological spaces.

In 1986, D. Andrijevic had defined and studied the concepts of semipreopen sets and semipreclosed sets in topology. In 2002, Navalagi has defined and studied classes of various continuous functions, open functions and closed functions in between topological spaces using semipreopen sets and semipreclosed sets. In this paper, we define and study some new classes of functions called p-semipreclosed functions and contra-p-semipreopen functions using semipreopen sets, preopen sets. Also, we characterize their basic properties. 2000 MSC: 54 A05, 54 C08 Keywords and Phrases: preopen sets, semipreopen sets, preopen functions, pre-semiopen functions, p-semipreopen functions.

International Journal of Engineering Sciences & Research Technology, 2013

In this paper by means of Bc open 

In this paper, we introduce the concept of $\alpha_{[\gamma, \gamma^{'}]}$-open sets in topological spaces and study some of their properties. Furthermore, we offer a new class of functions called $(\alpha_{[\gamma, \gamma^{'}]}$, $\alpha_{[\beta, \beta^{'}]})$-continuous functions and investigate their fundamental properties.

International Journal of Mathematics and Mathematical Sciences, 2001

We define a functionf:X→Yto be slightlyβ-continuous if for every clopen setVofY,f−1(V)⊂Cl(Int(Cl(f−1(V)))). We obtain several properties of such a function. Especially, we define the notion of ultra-regularizations of a topology and obtain interesting characterizations of slightlyβ-continuous functions by using it.