Continuity for Multifunctions in Ideal Topological Spaces

A new class of continuous multifunctions between an ideal topological space and a bitopological space, called upper (lower) $\tau^\star\alpha(\sigma_1,\sigma_2)$-continuous multifunctions, has been defined and studied. Moreover, several characterizations and some properties concerning upper $\tau^\star\alpha(\sigma_1,\sigma_2)$-continuous multifunctions and lower $\tau^\star\alpha(\sigma_1,\sigma_2)$-continuous multifunctions are established.

In this paper, a generalized class of ??*called weakly Ir?-open sets in ideal topological spaces is introduced and the notion of weakly Ir?-closed sets in ideal topological spaces is studied. The relationships of weakly Ir?-closed sets and various properties of weakly Ir?-closed sets are investigated.

In this paper we introduce and investigate some properties of semi*- I-open sets, pre*-I-open sets and e-I-open sets in ideal topological spaces. Moreover, some relationships among semi*-I-open sets, e-I-open sets and pre*-I-open sets in ideal topological spaces are established. Finally, we obtain the decompositions of continuity.

An ideal topological space is a triplet (X, τ, ℑ), where X is a nonempty set, τ is a topology on X, and ℑ is an ideal of subsets of X. In this paper, we introduce L∗-perfect, R∗-perfect, and C∗-perfect sets in ideal spaces and study their properties. We obtained a characterization for compatible ideals via R∗-perfect sets. Also, we obtain a generalized topology via ideals which is finer than τ using R∗-perfect sets on a finite set.

We introduce new classes of sets called a* -I -open,A-β-I-open sets, A-pre* -I-open sets, strongly T-I -sets, A-β-T-I-sets, strongly BA -I -sets, BA -I -sets, and δβA -I -open sets in ideal topological spaces. Using these sets, to obtain decompositions of continuity in an ideal topological space.

In this paper, we investigate some properties of a*-I-open set [3] and Asemi*-I-open sets defined in [3] in ideal topological space. The relationships of other related classes of sets are investigated. Also, a new decomposition of continuous functions is obtained by using δβA-I-open and SA*-set functions in an ideal topological space.

In this research we introduce h-local functions by using h-open sets in an ideal topological space (X, tau, I). Some properties and characterizations of the h-local functions are studied. Also, we introduce and research the notions of I_{{s^*}g}-h-closed and I_{g}-h-closed sets in an ideal topological space. Additionally, Cl^*_h is defined and its properties are examined.

Recently, Abbas has introduced and investigated the notion of h-open sets. In this paper, to generalize h-open sets, we introduce hI-open sets in an ideal topological space, and we obtain some properties of hI-open sets. We also introduce and investigate hI-continuous functions in an ideal topological space.

In this paper, we introduce generalized open sets, namely [Formula: see text]-pre-I-open, [Formula: see text]-semi-I-open, and [Formula: see text]-b-I-open sets in ideal topological spaces. We explain these notions thoroughly in the light of examples. Then, we give some of their properties and explore the relationships between these generalized open sets. Moreover, we define the [Formula: see text]-b-I-Hausdorff space in ideal topological spaces.

In this paper, we define new classes of sets called preopen sets, θI-semi-open sets, θI-β-open sets and θI-α-open sets in ideal topological spaces. Also, by using these sets, we obtain new decompositions of continuity in ideal topological spaces.

In this paper, we introduce and study the concepts upper (lower) almost -continuous multifunctions between topological spaces.

In this paper, by using β-open sets in [1] we introduce and investigate the conceptsof the β-local function, Is∗g-β-closed sets and Ig-β-closed sets in an ideal topological space. In addition to the properties, an operation cl∗β is defined and the properties are obtained similarly with the local function in [8].

A new class of closed sets, known as semi-regular weekly closed sets, is introduced in ideal topological space. In addition, we define a new class of continuous functions known as semi-regular weekly -continuous functions is introduced and studied. Some of their characteristics have also been examined and discussed interesting properties of semi-regular weekly -irresolute functions, strongly semi-regular weekly -continuous functions, and semi- regular weekly -continuous functions.

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.

In this paper we consider three types of space, and in these spaces, new types of generalized open set will be introduced. Further we shall introduce several types of topological space, and characterize the same.