Applying Set Theory E88

We prove some results in set theory as applied to general topology and model theory. In particular, we study ℵ1-collectionwise Hausdorff, Chang Conjecture for logics with Malitz-Magidor quantifiers and monadic logic of the real line by odd/even Cantor sets.

Complete Hausdorffness and Complete Regularity on Supra Topological Spaces

The supra topological topic is of great importance in preserving some topological properties under conditions weaker than topology and constructing a suitable framework to describe many real-life problems. Herein, we introduce the version of complete Hausdorffness and complete regularity on supra topological spaces and discuss their fundamental properties. We show the relationships between them with the help of examples. In general, we study them in terms of hereditary and topological properties and prove that they are closed under the finite product space. One of the issues we are interested in is showing the easiness and diversity of constructing examples that satisfy supra T i spaces compared with their counterparts on general topology.

La linguistique peut-elle sortir de son état pré-galiléen ?

The future of linguistics implies a better definition of concepts, especially in the semantic analysis. The notion of operator plays an important role in several areas of linguistics, for instance categorical grammars and representations of the meanings of grammatical categories. The general topology makes it possible to mathematize the grammatical concepts (time, aspects, modalities, enunciative operations) by means of operators. Curry’s Combinatorial Logic is an adequate formalism for composing and transforming operators at different levels of analysis that connect the semiotic expressions of languages (the observables) with their semantico-cognitive interpretations. The article refers to many studies that develop the points discussed.

Robust Formation Maintenance Methods under General Topology Pursuit of Multi-Agents Systems

In this article, methods of formation maintenance for a group of autonomous agents under ageneral topology scheme are discussed. Unlike rendezvous or geometric formation, general topology pursuit allows the group of agents to autonomously form trochoid patterns, which are useful in civilian and military applications. However, this type of topology is established by designing a marginally stable system that may be sensitive to parameter variations. To account for this drawback of stability, linear fixed-gains are turned into a dynamical version in this paper. By implementing a disturbance observer controller, systems are shown to maintain their formation despite the disturbances or uncertainties. Comparison in the effectiveness of the presented method with model reference adaptive control and integral sliding mode control under the uncertainties of the gains is also conducted. The capabilities of controllers are demonstrated and supported through simulations.

Infra -α- Compact and Infra -α- Connected Spaces

In 2016 Hakeem A. Othman and Md. Hanif Page introduced a new notion of set in general topology called an infra -α- open set and investigated its fundamental properties and studied the relationship between infra -α- open set and other topological sets. The objective of this paper is to introduce the new concepts called infra -α- compact space, countably infra -α- compact space, infra -α- Lindelof space, almost infra -α- compact space, mildly infra -α- compact space and infra -α- connected space in general topology and investigate several properties and characterizations of these new concepts in topological spaces.

Soft ideal topological spaces

The article deals with the correspondence between soft ideal topological spaces and ideal topological ones. Investigation of soft ideal topological spaces is based on methods of general topology and the application of results for soft omega open and strongly soft omega open sets is given.

Functionally Separation Axioms on General Topology

In this paper, we define a new family of separation axioms in the classical topology called functionally T i spaces for i = 0,1,2 . With the assistant of illustrative examples, we reveal the relationships between them as well as their relationship with T i spaces for i = 0,1,2 . We demonstrate that functionally T i spaces are preserved under product spaces, and they are topological and hereditary properties. Moreover, we show that the class of each one of them represents a transitive relation and obtain some interesting results under some conditions such as discrete and Sierpinski spaces.

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

In 2014 Mubarki, Al-Rshudi, and Al- Juhani introduced and studied the notion of a set in general topology called β*-open set and investigated its fundamental properties and studied the relationships between β*-open set and other topological sets including β*-continuity in topological spaces. We introduce and investigate several properties and characterizations of a new class of functions between topological spaces called β*- open, β*- closed, β*- continuous and β*- irresolute functions in topological spaces. We also introduce slightly β*- continuous, totally β*- continuous and almost β*- continuous functions between topological spaces and establish several characterizations of these new forms of functions. Furthermore, we also introduce and investigate certain ramifications of contra continuous and allied functions, namely, contra β*- continuous, and almost contra β*-continuous functions along with their several properties, characterizations and natural relationships. Moreover, we introduce new types of closed graphs by using β*- open sets and investigate its properties and characterizations in topological spaces.

Bi-interior Ideal Elements in ∧e-semigroups

All the results on semigroups obtained using only sets, can be written in an abstract form in a more general setting. Let us consider a recent paper to justify what we say. The bi-interior ideals of semigroups introduced and studied by M. Murali Krishna Rao in Discuss. Math. Gen. Algebra Appl. in 2018, follow for more general statements about ordered semigroups. The same holds for every result of this sort on semigroups based on right (left) ideals, bi-ideals, quasi-ideals, interior ideals etc. for which we use sets. As a result, we have an abstract formulation of the results on semigroups obtained by sets that is in the same spirit with the abstract formulation of general topology (the so-called topology without points) initiated by Koutsk ́y, Nöbeling and, even earlier, by Chittenden, Terasaka, Nakamura, Monteiro and Ribeiro. As a consequence, results on ordered Γ-hypersemigroups and on similar simpler structures can be obtained.