P-Normality

A topological space X is called P-normal if there exist a normal space Y and a bijective function f : X −→ Y such that the restriction f|A : A −→ f(A) is a homeomorphism for each paracompact subspace A ⊆ X. We will investigate this property and produce some examples to illustrate the relation between P-normality and other weaker kinds of normality.

On r -Generalized Fuzzy ℓ -Closed Sets: Properties and Applications

In the present study, we introduce and characterize the class of r -generalized fuzzy ℓ -closed sets in a fuzzy ideal topological space X , τ , ℓ in Šostak sense. Also, we show that r -generalized fuzzy closed set by Kim and Park (2002) ⟹ r -generalized fuzzy ℓ -closed set, but the converse need not be true. Moreover, if we take ℓ = ℓ 0 , the r -generalized fuzzy ℓ -closed set and r -generalized fuzzy closed set are equivalent. After that, we define fuzzy upper (lower) generalized ℓ -continuous multifunctions, and some properties of these multifunctions along with their mutual relationships are studied with the help of examples. Finally, some separation axioms of r -generalized fuzzy ℓ -closed sets are introduced and studied. Also, the notion of r -fuzzy G ∗ -connected sets is defined and studied with help of r -generalized fuzzy ℓ -closed sets.

Electrostatic-Elastic MEMS with Fringing Field: A Problem of Global Existence

In this paper, we prove the existence and uniqueness of solutions for a nonlocal, fourth-order integro-differential equation that models electrostatic MEMS with parallel metallic plates by exploiting a well-known implicit function theorem on the topological space framework. As the diameter of the domain is fairly small (similar to the length of the device wafer, which is comparable to the distance between the plates), the fringing field phenomenon can arise. Therefore, based on the Pelesko–Driscoll theory, a term for the fringing field has been considered. The nonlocal model obtained admits solutions, making these devices attractive for industrial applications whose intended uses require reduced external voltages.

Neutrosophic Orbit Topological Spaces

Neutrosophy is a flourishing arena which conceptualizes the notion of true, falsity and indeterminancy attributes of an event. In the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. Hence in this paper we focus on introducing the concept of neutrosophic orbit topological space denoted as (X, tNO). Also, some of the important characteristics of neutrosophic orbit open sets are discussed with suitable examples. HIGHLIGHTS The orbit in mathematics has an important role in the study of dynamical systems Neutrosophy is a flourishing arena which conceptualizes the notion of true, falsity and indeterminancy attributes of an event. We combine the above two topics and create the following new concept The collection of all neutrosophic orbit open sets under the mapping . we introduce the necessary conditions on the mapping 𝒇 in order to obtain a fixed orbit of a neutrosophic set (i.e., 𝒇(𝝁) = 𝝁) for any neutrosophic orbit open set 𝝁 under the mapping 𝒇

A study on compactness in intuitionistic fuzzy topological spaces

The purpose of this paper is to introduce and study the compactness in intuitionistic fuzzy topological spaces. Here we define two new notions of intuitionistic fuzzy compactness in intuitionistic fuzzy topological space and find their relation. Also we find the relationship between intuitionistic general compactness and intuitionistic fuzzy compactness. Here we see that our notions satisfy hereditary and productive property.

Topological Essence of the Concept “Limit of a Function” in the General Mathematics

The limit of a function is one of the most important concepts in high school mathematics. But unfortunately, not many students understand the essence of this concept. They had to accept it. This paper presents the topological essence of a limit of a function and discusses methods for teaching this concept in general education in Viet Nam. It consists of four parts. Part 1 talks about the definition of “Limit of a function”, part 2 covers the Mathematical essence of the concept limit of a function, part 3 deals with Defining the limit of a function in a topological space of real numbers and part 4 presents Adding a topology on the set of real numbers and limit of a function. In each section, we include comments to help teach these notions and concepts.

Decomposition, Mapping, and Sum Theorems of ω-Paracompact Topological Spaces

As a weaker form of ω-paracompactness, the notion of σ-ω-paracompactness is introduced. Furthermore, as a weaker form of σ-ω-paracompactness, the notion of feebly ω-paracompactness is introduced. It is proven hereinthat locally countable topological spaces are feebly ω-paracompact. Furthermore, it is proven hereinthat countably ω-paracompact σ-ω-paracompact topological spaces are ω-paracompact. Furthermore, it is proven hereinthat σ-ω-paracompactness is inverse invariant under perfect mappings with countable fibers, and as a result, is proven hereinthat ω-paracompactness is inverse invariant under perfect mappings with countable fibers. Furthermore, if A is a locally finite closed covering of a topological space X,τ with each A∈A being ω-paracompact and normal, then X,τ is ω-paracompact and normal, and as a corollary, a sum theorem for ω-paracompact normal topological spaces follows. Moreover, three open questions are raised.

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.

Force-Mediating Particle of Coupling of Spin Angular Momenta

In this paper, a hypothesis is proposed, that something similar to what happen to the puzzle of the energy losing in decay of neutron may also occur to the puzzle of the sum losing of the z-components of spin angular momenta in the synthetic course of spin coupling in Spin Topological Space. The former puzzle is related to hidden neutrial antineutrino that carries a small amount of energy away, the latter puzzle is related to hidden "constructive" zero-spin particle playing the role of a force-mediator that carries some amount of spin angular momentum, which just offsets the same amount of angular momentum losing in the formation of spin coupling.