Topological Structures via Interval-Valued Neutrosophic Crisp Sets

In this paper, we introduce the new notion of interval-valued neutrosophic crisp sets providing a tool for approximating undefinable or complex concepts in real world. First, we deal with some of its algebraic structures. We also define an interval-valued neutrosophic crisp (vanishing) point and obtain some of its properties. Next, we define an interval-valued neutrosophic crisp topology, base (subbase), neighborhood, and interior (closure), respectively and investigate some of each property, and give some examples. Finally, we define an interval-valued neutrosophic crisp continuity and quotient topology and study some of each property.

New topology in residuated lattices

In this paper, by using the notion of upsets in residuated lattices and defining the operator Da(X), for an upset X of a residuated lattice L we construct a new topology denoted by τa and (L, τa) becomes a topological space. We obtain some of the topological aspects of these structures such as connectivity and compactness. We study the properties of upsets in residuated lattices and we establish the relationship between them and filters. O. Zahiri and R. A. Borzooei studied upsets in the case of BL-algebras, their results become particular cases of our theory, many of them work in residuated lattices and for that we offer complete proofs. Moreover, we investigate some properties of the quotient topology on residuated lattices and some classes of semitopological residuated lattices. We give the relationship between two types of quotient topologies τa/F and $\begin{array}{} \displaystyle \mathop {{\tau _a}}\limits^ - \end{array}$. Finally, we study the uniform topology $\begin{array}{} \displaystyle {\tau _{\bar \Lambda }} \end{array}$ and we obtain some conditions under which $\begin{array}{} \displaystyle (L/J,{\tau _{\bar \Lambda }}) \end{array}$ is a Hausdorff space, a discrete space or a regular space ralative to the uniform topology. We discuss briefly the applications of our results on classes of residuated lattices such as divisible residuated lattices, MV-algebras and involutive residuated lattices and we find that any of this subclasses of residuated lattices with respect to these topologies form semitopological algebras.

Metric on the spectrum of the algebra of entire symmetric functions of bounded type on the complex $L_\infty$

It is known that every complex-valued homomorphism of the Fréchet algebra $H_{bs}(L_\infty)$ of all entire symmetric functions of bounded type on the complex Banach space $L_\infty$ is a point-evaluation functional $\delta_x$ (defined by $\delta_x(f) = f(x)$ for $f \in H_{bs}(L_\infty)$) at some point $x \in L_\infty.$ Therefore, the spectrum (the set of all continuous complex-valued homomorphisms) $M_{bs}$ of the algebra $H_{bs}(L_\infty)$ is one-to-one with the quotient set $L_\infty/_\sim,$ where an equivalence relation "$\sim$'' on $L_\infty$ is defined by $x\sim y \Leftrightarrow \delta_x = \delta_y.$ Consequently, $M_{bs}$ can be endowed with the quotient topology. On the other hand, $M_{bs}$ has a natural representation as a set of sequences which endowed with the coordinate-wise addition and the quotient topology forms an Abelian topological group. We show that the topology on $M_{bs}$ is metrizable and it is induced by the metric $d(\xi, \eta) = \sup_{n\in\mathbb{N}}\sqrt[n]{|\xi_n-\eta_n|},$ where $\xi = \{\xi_n\}_{n=1}^\infty,\eta = \{\eta_n\}_{n=1}^\infty \in M_{bs}.$

Topology on the spectrum of the algebra of entire symmetric functions of bounded type on the complex $L_\infty$

It is known that the so-called elementary symmetric polynomials $R_n(x) = \int_{[0,1]}(x(t))^n\,dt$ form an algebraic basis in the algebra of all symmetric continuous polynomials on the complex Banach space $L_\infty,$ which is dense in the Fr\'{e}chet algebra $H_{bs}(L_\infty)$ of all entire symmetric functions of bounded type on $L_\infty.$ Consequently, every continuous homomorphism $\varphi: H_{bs}(L_\infty) \to \mathbb{C}$ is uniquely determined by the sequence $\{\varphi(R_n)\}_{n=1}^\infty.$ By the continuity of the homomorphism $\varphi,$ the sequence $\{\sqrt[n]{|\varphi(R_n)|}\}_{n=1}^\infty$ is bounded. On the other hand, for every sequence $\{\xi_n\}_{n=1}^\infty \subset \mathbb{C},$ such that the sequence $\{\sqrt[n]{|\xi_n|}\}_{n=1}^\infty$ is bounded, there exists $x_\xi \in L_\infty$ such that $R_n(x_\xi) = \xi_n$ for every $n \in \mathbb{N}.$ Therefore, for the point-evaluation functional $\delta_{x_\xi}$ we have $\delta_{x_\xi}(R_n) = \xi_n$ for every $n \in \mathbb{N}.$ Thus, every continuous complex-valued homomorphism of $H_{bs}(L_\infty)$ is a point-evaluation functional at some point of $L_\infty.$ Note that such a point is not unique. We can consider an equivalence relation on $L_\infty,$ defined by $x\sim y \Leftrightarrow \delta_x = \delta_y.$ The spectrum (the set of all continuous complex-valued homomorphisms) $M_{bs}$ of the algebra $H_{bs}(L_\infty)$ is one-to-one with the quotient set $L_\infty/_\sim.$ Consequently, $M_{bs}$ can be endowed with the quotient topology. On the other hand, it is naturally to identify $M_{bs}$ with the set of all sequences $\{\xi_n\}_{n=1}^\infty \subset \mathbb{C}$ such that the sequence $\{\sqrt[n]{|\xi_n|}\}_{n=1}^\infty$ is bounded.We show that the quotient topology is Hausdorffand that $M_{bs}$ with the operation of coordinate-wise addition of sequences forms an abelian topological group.

A Topology on Milnor’s Group of a Topological Field and Continuous Joint Determinants

For the tuple set of commuting invertible matrices with coefficients in a given field, the joint determinants are defined as generalizations of the determinant map for the square matrices. We introduce a natural topology on Milnor’s K-groups of a topological field as the quotient topology induced by the joint determinant map and investigate the existence of a nontrivial continuous joint determinant by utilizing this topology, generalizing the author’s previous results on the continuous joint determinants for the commuting invertible matrices over R and C.

Slice Theorem For Differential Spaces And Reduction By Stages

We show that the space P/G of orbits of a proper action of a Lie group G on a locally compact differential space P is a locally compact differential space with quotient topology. Applying this result to reduction of symmetries of Hamiltonian systems, we prove the reduction by stages theorem

Topological structuring of the digital plane

Combinatorics International audience We discuss an Alexandroff topology on ℤ2 having the property that its quotient topologies include the Khalimsky and Marcus-Wyse topologies. We introduce a further quotient topology and prove a Jordan curve theorem for it.