Soft Semi ω-Open Sets

In this paper, we introduce the class of soft semi ω-open sets of a soft topological space (X,τ,A), using soft ω-open sets. We show that the class of soft semi ω-open sets contains both the soft topology τω and the class of soft semi-open sets. Additionally, we define soft semi ω-closed sets as the class of soft complements of soft semi ω-open sets. We present here a study of the properties of soft semi ω-open sets, especially in (X,τ,A) and (X,τω,A). In particular, we prove that the class of soft semi ω-open sets is closed under arbitrary soft union but not closed under finite soft intersections; we also study the correspondence between the soft topology of soft semi ω-open sets of a soft topological space and their generated topological spaces and vice versa. In addition to these, we introduce the soft semi ω-interior and soft semi ω-closure operators via soft semi ω-open and soft semi ω-closed sets. We prove several equations regarding these two new soft operators. In particular, we prove that these operators can be calculated using other usual soft operators in both of (X,τ,A) and (X,τω,A), and some equations focus on soft anti-locally countable soft topological spaces.

Applications of operations on generalized topological spaces

In this paper γµ -open sets and γµ -closed sets in a GTS (X, µ) have been studied, where γµ is an operation from µ to P(X). In general, collection of γµ -open sets is smaller than the collection of µ-open sets. The condition under which both are same are also established here. Some properties of such sets have been discussed. Some closure like operators are also defined and their properties are discussed. The relation between similar types of closure operators on the GTS (X, µ) has been established. The condition under which the newly defined closure like operator is a Kuratowski closure operator is given. We have also defined a generalized type of closed sets termed as γµ -generalized closed set with the help of this newly defined closure operator and discussed some basic properties of such sets. As an application, we have introduced some weak separation axioms and discussed some of their properties. Finally, we have shown some preservation theorems of such generalized concepts.

Soft Continuous Mappings in Soft Closure Spaces

Soft closure spaces are a new structure that was introduced very recently. These new spaces are based on the notion of soft closure operators. This work aims to provide applications of soft closure operators. We introduce the concept of soft continuous mappings and soft closed (resp. open) mappings, support them with examples, and investigate some of their properties.

On Single-Valued Neutrosophic Closure Spaces

This paper aims to mark out new terms of single-valued neutrosophic notions in a Šostak sense called single-valued neutrosophic semi-closure spaces. To achieve this, notions such as β£-closure operators and β£-interior operators are first defined. More precisely, these proposed contributions involve different terms of single-valued neutrosophic continuous mappings called single-valued neutrosophic (almost β£, faintly β£, weakly β£) and β£-continuous. Finally, for the purpose of symmetry, we define the single-valued neutrosophic upper, single-valued neutrosophic lower and single-valued neutrosophic boundary sets of a rough single-valued neutrosophic set αn in a single-valued neutrosophic approximation space (F˜,δ). Based on αn and δ, we also introduce the single-valued neutrosophic approximation interior operator intαnδ and the single-valued neutrosophic approximation closure operator Clαnδ.

New Soft Structure: Infra Soft Topological Spaces

It is always convenient to find the weakest conditions that preserve some topologically inspired properties. To this end, we introduce the concept of an infra soft topology which is a collection of subsets that extend the concept of soft topology by dispensing with the postulate that the collection is closed under arbitrary unions. We study the basic concepts of infra soft topological spaces such as infra soft open and infra soft closed sets, infra soft interior and infra soft closure operators, and infra soft limit and infra soft boundary points of a soft set. We reveal the main properties of these concepts with the help of some elucidative examples. Then, we present some methods to generate infra soft topologies such as infra soft neighbourhood systems, basis of infra soft topology, and infra soft relative topology. We also investigate how we initiate an infra soft topology from crisp infra topologies. In the end, we explore the concept of continuity between infra soft topological spaces and determine the conditions under which the continuity is preserved between infra soft topological space and its parametric infra topological spaces.

The orbit of closure-involution operations: the case of Boolean functions

For a set A of Boolean functions, a closure operator c and an involution i, let $$\mathcal{N}_{c,i}(A)$$ N c , i ( A ) be the number of sets which can be obtained from A by repeated applications of c and i. The orbit $$\mathcal{O}(c,i)$$ O ( c , i ) is defined as the set of all these numbers. We determine the orbits $$\mathcal{O}(S,i)$$ O ( S , i ) where S is the closure defined by superposition and i is the complement or the duality. For the negation $${{\,\mathrm{non}\,}}$$ non , the orbit $$\mathcal{O}(S,{{\,\mathrm{non}\,}})$$ O ( S , non ) is almost determined. Especially, we show that the orbit in all these cases contains at most seven numbers. Moreover, we present some closure operators where the orbit with respect to duality and negation is arbitrarily large.

Onto extensions of free groups

An extension of subgroups $H\leqslant K\leqslant F_A$ of the free group of rank $|A|=r\geqslant 2$ is called onto when, for every ambient free basis $A'$, the Stallings graph $\Gamma_{A'}(K)$ is a quotient of $\Gamma_{A'}(H)$. Algebraic extensions are onto and the converse implication was conjectured by Miasnikov-Ventura-Weil, and resolved in the negative, first by Parzanchevski-Puder for rank $r=2$, and recently by Kolodner for general rank. In this note we study properties of this new type of extension among free groups (as well as the fully onto variant), and investigate their corresponding closure operators. Interestingly, the natural attempt for a dual notion -- into extensions -- becomes trivial, making a Takahasi type theorem not possible in this setting.

e#-open and *e-open sets via -open sets

The notion of -open sets in a topological space was studied by Velicko. Usha Parmeshwari et.al. and Indira et.al. introduced the concepts of b# and *b open sets respectively. Following this Ekici et. al. studied the notions of e-open and e-closed sets by mixing the closure, interior, -interior and -closure operators. In this paper some new sets namely e#-open and *e-open sets are defined and their relationship with other similar concetps in topological spaces will be investigated.

On L-fuzzy closure operators and L-fuzzy pre-proximities

The aim of this paper is to investigate the relations among the L-fuzzy pre-proximities, L-fuzzy closure operators and L-fuzzy co-topologies in complete residuated lattices. We show that there is a Galois correspondence between the category of separated L-fuzzy closure spaces and that of separated L-fuzzy pre-proximity spaces and we give their examples.