Fuzzy Closure Hopf Space

This paper studies quasi-discrete closure spaces and fuzzy closure spaces. We show that any topological closure cT induced by a closure c is the smallest extension from a closure space to a topological closure space in both crisp and fuzzy environment, in addition, a characterization of the continuous mappings in quasi-discrete closure spaces is obtained. We propose the concept of quasi-discrete fuzzy closure spaces in the context of fuzzy sets and establish a one to one correspondence between quasi-discrete fuzzy closure spaces and reflexive fuzzy relations. We also discuss the relationship between topological closure cT and closure c in quasi-discrete fuzzy closure spaces and show that the process from closure c to topological closure cT can be realized via the process from a reflexive fuzzy relation to its transitive closure.

The aim of this paper is to introduce $(L,M)$-fuzzy closurestructure where $L$ and $M$ are strictly two-sided, commutativequantales. Firstly, we define $(L,M)$-fuzzy closure spaces and getsome relations between $(L,M)$-double fuzzy topological spaces and$(L,M)$-fuzzy closure spaces. Then, we introduce initial$(L,M)$-fuzzy closure structures and we prove that the category$(L,M)$-{bf FC} of $(L,M)$-fuzzy closure spaces and$(L,M)$-$mathcal{C}$-maps is a topological category over thecategory {bf SET}. From this fact, we define products of$(L,M)$-fuzzy closure spaces. Finally, we show that an initialstructure of $(L,M)$-double fuzzy topological spaces can be obtainedby the initial structure of $(L,M)$-fuzzy closure spaces induced bythem.

A fuzzy ?ech closure space (X, k) is a fuzzy set X with fuzzy ?ech closure operator k: IX ? IX where IX is a power set of fuzzy subsets of X, which satisfies k ( ) = , 1 ?2 ? k( 1 ) k( ?2 ), k ( 1 ?2 ) = k 1) ?k (?2) for all 1 , ?2 IX . A fuzzy topological space X is said to be fuzzy connected if it has no proper fuzzy clopen set.Many properties which hold in fuzzy topological space hold in fuzzy ?ech closure space as well. A ?ech closure space (X, u) is said to be connected if and only if any continuous map f from X to the discrete space {0, 1} is constant. In this paper we introduce connectedness in fuzzy ?ech closure space. Keywords: Connectedness in Fuzzy ?ech Closure Space, Connectedness in Fuzzy Topological Space, Fuzzy ?ech Closure Operator, Fuzzy ?ech Closure Space, Fuzzy Topological Space.

Given a homotopy equivalence f : X → Y f:X \to Y , a homotopy inverse g of f, and a homotopy H : X × I → X H:X \times I \to X from g ∘ f g \circ f to 1 X {1_X} . We show that there is a homotopy K : Y × I → Y K:Y \times I \to Y from f ∘ g f \circ g to 1 Y {1_Y} such that f ∘ H ≃ K ∘ ( f × 1 I ) rel X × ∂ I f \circ H \simeq K \circ (f \times {1_I})\,{\text {rel}}\,X \times \partial I and H ∘ ( g × 1 I ) ≃ g ∘ K rel Y × ∂ I H \circ (g \times {1_I}) \simeq g \circ K\,{\text {rel}}\,Y \times \partial I .

Metric spaces and their various generalizations occur frequently in computer science applications. This is the reason why, in this paper, we introduced and studied the concept of fuzzy b-metric space, generalizing, in this way, both the notion of fuzzy metric space introduced by I. Kramosil and J. Michálek and the concept of b-metric space. On the other hand, we introduced the concept of fuzzy quasi-bmetric space, extending the notion of fuzzy quasi metric space recently introduced by V. Gregori and S. Romaguera. Finally, a decomposition theorem for a fuzzy quasipseudo- b-metric into an ascending family of quasi-pseudo-b-metrics is established. The use of fuzzy b-metric spaces and fuzzy quasi-b-metric spaces in the study of denotational semantics and their applications in control theory will be an important next step.

In this paper, we join the notion of fuzzy ideal to the notion of fuzzy approximation space to define the notion of fuzzy ideal approximation spaces. We introduce the fuzzy ideal approximation interior operator int Φ λ and the fuzzy ideal approximation closure operator cl Φ λ , and moreover, we define the fuzzy ideal approximation preinterior operator p int Φ λ and the fuzzy ideal approximation preclosure operator p cl Φ λ with respect to that fuzzy ideal defined on the fuzzy approximation space (X, R) associated with some fuzzy set λ ∈ IX. Also, we define fuzzy separation axioms, fuzzy connectedness and fuzzy compactness in fuzzy approximation spaces and in fuzzy ideal approximation spaces as well, and prove the implications in between.

Fuzzy linear spaces

This book introduces a new context for global homotopy theory, i.e., equivariant homotopy theory with universal symmetries. Many important equivariant theories naturally exist not just for a particular group, but in a uniform way for all groups in a specific class. Prominent examples are equivariant stable homotopy, equivariant $K$-theory or equivariant bordism. Global equivariant homotopy theory studies such uniform phenomena, i.e., the adjective `global' refers to simultaneous and compatible actions of all compact Lie groups. We give a self-contained treatment of unstable and stable global homotopy theory, modeled by orthogonal spaces respectively orthogonal spectra under global equivalences. Specific topics include the global stable homotopy category, operations on equivariant homotopy groups, global model structures, and ultra-commutative multiplications. The book includes many explicit examples and detailed calculations.

Homotopy theory and simplicial groupoids

Since we have removed 0 + jO from the complex plane, the Nyquist map D × Ω (ℂ \ {0 + j0}, if it exists, could have a nontrivial homotopy class. The set of all possible homotopy classes of all such maps depends essentially on D. This chapter addresses the question as to what homotopy classes can be expected for a given uncertainty space. In case of such uncontractible uncertainty spaces as toris—multichannel phase margin problems—the set of all homotopy classes is quite large. The next question is to determine which of these homotopy classes yield closed-loop stability. Finally, given an uncertainty structure D, it is shown how the loop function L(s) induces a homotopy class of the map.

Our goal in the present paper is to recall the concept of general fuzzy normed space and its basic properties to define the general fuzzy bounded operator as a background to introduce the notion general fuzzy norm of a general fuzzy bounded linear operator. After that we proved any operator from a general fuzzy normed space into a general complete general fuzzy normed space has an extension. Also we prove that a general fuzzy bounded operator on a general fuzzy normed space is equivalent to a general fuzzy continuous. Finally different types of fuzzy approaches of operators is introduced in order to prove that the general fuzzy normed space GFB(V,U) is general complete when U is general complete.

Let M be a compact connected Riemannian manifold whose sectional curvature values are all nonpositive. Let T denote the fundamental group of M. We prove that any homotopy equivalence : N -> M from a compact closed manifold TV is homotopic to a homeomorphism, provided that m > 5 where m = dim M . We show that the surgery L-group L k+m (T, w { ) is isomorphic to the set of homotopy classes of k k

In this paper, the C̆ech fuzzy soft closure spaces are defined and their basic properties are studied. Closed (respectively, open) fuzzy soft sets is defined in C̆ech fuzzy-soft closure spaces. It has been shown that for each C̆ech fuzzy soft closure space there is an associated fuzzy soft topological space. In addition, the concepts of a subspace and a sum are defined in C̆ech fuzzy soft closure space. Finally, fuzzy soft continuous (respectively, open and closed) mapping between C̆ech fuzzy soft closure spaces are introduced. Mathematics Subject Classification: 54A40, 54B05, 54C05.

In this paper, the concepts of somewhat fuzzy automata continuous functions and somewhat fuzzy automata open functions in fuzzy automata topological spaces are introduced and some interesting properties of these functions are studied. In this connection, the concepts of fuzzy automata resolvable spaces and fuzzy automata irresolvable spaces are also introduced and their properties are studied.

This paper explores fundamental applications of topological spaces and stereographic projection derived from the properties of the circle, employing key concepts such as continuous functions and homotopy theory. By examining the behavior of mappings and deformations within topological spaces, the study demonstrates how the circle serves as a foundational structure for understanding more complex topological constructs. Special attention is given to the use of stereographic projection in visualizing the relationship between the circle and the unit sphere, illustrating how these mathematical tools contribute to a deeper understanding of continuity and homotopy in topological analysis. The discussion offers a concise yet insightful introduction to the interplay between geometric intuition and topological formalism.