Baire Category Theorem Research Articles

On Neutrosophic Fuzzy Metric Space and Its Topological Properties

The present research introduces a novel concept termed “neutrosophic fuzzy metric space”, which extends the traditional metric space framework by incorporating the notion of neutrosophic fuzzy sets. A thorough investigation of various structural and topological properties within this newly proposed generalization of metric space has been conducted. Additionally, counterparts of well-known theorems such as the Uniform Convergence Theorem and the Baire Category Theorem have been established for this generalized metric space. Through rigorous analysis, a detailed understanding of its fundamental characteristics has been attained, illuminating its potential applications and theoretical significance.

On Non-Archimedean $\mathcal{L}$-Fuzzy Vector Metric Spaces

This paper contributes to the broader studies of fuzzy vector metric spaces and fuzzy metric spaces based on order structures beyond the unit interval. It defines the notions of the left (right) order convergence and continuity in non-Arcimedean $\mathcal{L}$-fuzzy vector metric spaces. The notation $\mathcal{M}_E(a,b,s)$ means the nearness between $a$ and $b$ according to any positive vector $s$. This study exemplifies definitions and reaches some well-known results. Moreover, it proposes the concept of $\mathcal{L}$-fuzzy vector metric diameter and studies some of its basic properties. Further, the present paper proves the Cantor intersection theorem and the Baire category theorem via these concepts. Finally, this study discusses the need for further research.

Generic algebraic properties in spaces of enumerated groups

We introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using elementary tools and well-known examples from combinatorial group theory, combined with the Baire category theorem, we obtain a plethora of results demonstrating that several phenomena in group theory are generic. In effect, we provide a new topological framework for the analysis of various well known problems in group theory. We also provide a connection between genericity in these spaces, the word problem for finitely generated groups and model-theoretic forcing. Using these connections, we investigate a natural question raised by Osin: when does a certain space of enumerated groups contain a comeager isomorphism class? We obtain a sufficient condition that allows us to answer Osin’s question in the negative for the space of all enumerated groups and the space of left orderable enumerated groups. We document several open questions in connection with these considerations.

A generalization of the Open Mapping Theorem and a possible generalization of the Baire Category Theorem

We prove a generalization of the Open Mapping Theorem for locally compact σ-compact groups in which we replace the assumption of the domain being σ-compact with a weaker condition. We provide a characterisation of the cardinal number c and we suggest a possibility of generalizing the Baire Category Theorem.

COMPARING COMPUTABILITY IN TWO TOPOLOGIES

Abstract Computable analysis provides ways of representing points in a topological space, and therefore of defining a notion of computable points of the space. In this article, we investigate when two topologies on the same space induce different sets of computable points. We first study a purely topological version of the problem, which is to understand when two topologies are not $\sigma $ -homeomorphic. We obtain a characterization leading to an effective version, and we prove that two topologies satisfying this condition induce different sets of computable points. Along the way, we propose an effective version of the Baire category theorem which captures the construction technique, and enables one to build points satisfying properties that are co-meager with respect to a topology, and are computable with respect to another topology. Finally, we generalize the result to three topologies and give an application to prove that certain sets do not have computable type, which means that they have a homeomorphic copy that is semicomputable but not computable.

The Baire theorem, an analogue of the Banach fixed point theorem and attractors in T1 compact spaces

We prove that if X is a T1 second countable compact space, then X is a Baire space if and only if every nonempty open subset of X contains a closed subset with nonempty interior. We also prove an analogue of Banach's fixed point theorem for all T1 compact spaces. Applying the analogue of Banach's fixed point theorem we prove the existence of unique attractors for so called contractive iterated function systems whose Hutchinson operators are closed in compact T1 spaces.

Common fixed point, Baire's and Cantor's theorems in neutrosophic 2-metric spaces

<abstract> <p>These fundamental Theorems of classical analysis, namely Baire's Theorem and Cantor's Intersection Theorem in the context of Neutrosophic 2-metric spaces, are demonstrated in this article. Naschie discussed high energy physics in relation to the Baire's Theorem and the Cantor space in descriptive set theory. We describe, how to demonstrate the validity and uniqueness of the common fixed-point theorem for four mappings in Neutrosophic 2-metric spaces.</p> </abstract>

Generalized harmonic functions on trees: Universality and frequent universality

Recently, harmonic functions and frequently universal harmonic functions on a tree T have been studied, taking values on a separable Fréchet space E over the field C or R. In the present paper, we allow the functions to take values in a vector space E over a rather general field F. The metric of the separable topological vector space E is translation invariant and instead of harmonic functions we can also study more general functions defined by linear combinations with coefficients in F. We don't assume that E is complete and therefore we present an argument avoiding Baire's theorem.

Index of symmetry and topological classification of asymmetric normed spaces

Let X,Y be asymmetric normed spaces and Lc(X,Y) the convex cone of all linear continuous operators from X to Y. It is known that in general, Lc(X,Y) is not a vector space. The aim of this note is to give, using the Baire category theorem, a complete characterization on X and a finite dimensional Y so that Lc(X,Y) is a vector space. For this, we introduce an index of symmetry of the space X denoted c(X)∈[0,1] and we give the link between the index c(X) and the fact that Lc(X,Y) is in turn an asymmetric normed space for every asymmetric normed space Y. Our study leads to a topological classification of asymmetric normed spaces.

Two applications of topology to model theory

By utilizing the topological concept of pseudocompactness, we simplify and improve a proof of Caicedo, Dueñez, and Iovino concerning Terence Tao's metastability. We also pinpoint the exact relationship between the Omitting Types Theorem and the Baire Category Theorem by developing a machine that turns topological spaces into abstract logics.

Baire theorem and hypercyclic algebras

The question of whether a hypercyclic operator T acting on a Fréchet algebra X admits or not an algebra of hypercyclic vectors (but 0) has been addressed in the recent literature. In this paper we give new criteria and characterisations in the context of convolution operators acting on H(C) and backward shifts acting on a general Fréchet sequence algebra. Analogous questions arise for stronger properties like frequent hypercyclicity. In this trend we give a sufficient condition for a weighted backward shift to admit an upper frequently hypercyclic algebra and we find a weighted backward shift acting on c0 admitting a frequently hypercyclic algebra for the coordinatewise product. The closed hypercyclic algebra problem is also covered.

Critical counterexamples for linear wave equations with time-dependent propagation speed

We investigate an abstract wave equation with a time-dependent propagation speed, and we consider both the non-dissipative case, and the case with a strong damping that depends on a power of the elastic operator. Previous results show that, depending on the values of the parameters and on the time regularity of the propagation speed, this equation exhibits either well-posedness in Sobolev spaces, or well-posedness in Gevrey spaces, or ill-posedness with severe derivative loss.In this paper we examine some critical cases that were left open by the previous literature, and we show that they fall into the pathological regime. The construction of the counterexamples requires a redesign from scratch of the basic ingredients, and a suitable application of Baire category theorem in place of the usual iteration scheme.

Neutrosophic metric spaces

Neutrosophy consists of neutrosophic logic, probability, and sets. Actually, the neutrosophic set is a generalisation of classical sets, fuzzy set, intuitionistic fuzzy set, etc. A neutrosophic set is a mathematical notion serving issues containing inconsistent, indeterminate, and imprecise data. The notion of intuitionistic fuzzy metric space is useful in modelling some phenomena where it is necessary to study the relationship between two probability functions. In this paper, the definition of new metric space with neutrosophic numbers is given. Neutrosophic metric space uses the idea of continuous triangular norms and continuous triangular conorms in intuitionistic fuzzy metric space. Triangular norms are used to generalize with the probability distribution of triangle inequality in metric space conditions. Triangular conorms are known as dual operations of triangular norms. Triangular norms and triangular conorm are very significant for fuzzy operations. Neutrosophic metric space was defined with continuous triangular norms and continuous triangular conorms. Several topological and structural properties neutrosophic metric space have been investigated. The analogues of Baire Category Theorem and Uniform Convergence Theorem are given for Neutrosophic metric spaces.

Stability of Mixed Additive–Quadratic and Additive–Drygas Functional Equations

In this paper, using the Baire category theorem we investigate the Hyers–Ulam stability problem of mixed additive–quadratic and additive–Drygas functional equations $$\begin{aligned} 2f(x+y) + f(x-y) - 3f(x) -3f(y)&= 0,\\ 2f(x+y) + f(x-y) - 3f(x) -2f(y) -f(-y)&= 0 \end{aligned}$$on a set of Lebesgue measure zero. As a consequence, we obtain asymptotic behaviors of the functional equations.

Nonhyperbolic Coxeter groups with Menger boundary

A generic finite presentation defines a word hyperbolic group whose boundary is homeomorphic to the Menger curve. In this article we produce the first known examples of non-hyperbolic CAT(0) groups whose visual boundary is homeomorphic to the Menger curve. The examples in question are the Coxeter groups whose nerve is a complete graph on n vertices for n \geq 5 . The construction depends on a slight extension of Sierpinski’s theorem on embedding 1-dimensional planar compacta into the Sierpinski carpet. We give a simplified proof of this theorem using the Baire category theorem.

Simple equivariant C*-algebras whose full and reduced crossed products coincide

For any second countable locally compact group G , we construct a simple G -C*- algebra whose full and reduced crossed product norms coincide. We then construct its G -equivariant representation on another simple G -C*-algebra without the coincidence condition. This settles two problems posed by Anantharaman-Delaroche in 2002. Some constructions involve the Baire category theorem.

Measure zero stability problem of a generalized quadratic functional equation

Let X be a normed space, Y be a Banach space and $$f,g: X\rightarrow Y$$. In this paper, we investigate the Hyers–Ulam stability theorem for the generalized quadratic functional equation $$\begin{aligned} f(kx+y)+f(kx-y)=2k^2g(x)+2f(y) \end{aligned}$$in a set $$\Omega \subset X\times X$$, where k is a positive integer. By the Baire category theorem, we derive some consequences of our main result.

Generalized Open Mapping Theorem for X-Normed Spaces

The theory of X-normed spaces over non-Archimedean valued fields with valuations of higher rank was introduced by H. Ochsenius and W. H. Schikhof in [9] and further developed in [10–12, 16, 17] and [13]. In order to obtain results like the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem, H. Ochsenius and W. H. Schikhof used 1st countability conditions in the value group of the based field. In this article the author develops a new tool to work with transfinite induction simplifying the techniques employed in X-normed spaces, thus accomplishing a Generalized Baire Category Theorem that allows the proof of an Open Mapping Theorem for X-normed spaces without restrictions on the value group of the based field. Additionally, some contributions to the theory of X-normed spaces are presented regarding quotient spaces.

A generalization of a Baire theorem concerning barely continuous functions

We prove that if X is a paracompact space, Y is a metric space and f:X→Y is a functionally fragmented map, then (i) f is σ-discrete and functionally Fσ-measurable; (ii) f is a Baire-one function, if Y is weak adhesive and weak locally adhesive for X; (iii) f is countably functionally fragmented, if X is Lindelöff.This result generalizes one theorem of René Baire on classification of barely continuous functions.

Linear Structure of Functions with Maximal Clarke Subdifferential

In this paper we establish that the set of Lipschitz functions $f:\mathcal{U}\rightarrow \mathbb{R}$ ($\mathcal{U}$ a nonempty open subset of $\ell_{d}^{1}$) with maximal Clarke subdifferential contains a linear subspace of uncountable dimension (in particular, an isometric copy of $\ell^{\infty}(\mathbb{N})$). This result follows along a similar line to that of a previous result of Borwein and Wang (see [Proc. Amer. Math. Soc., 128 (2000), pp. 3221--3229; Bull. Aust. Math. Soc., 72 (2005), pp. 491--496]). However, while the latter was based on Baire's category theorem, our current approach is constructive and is not linked to uniform convergence. In particular, we establish lineability (and spaceability for the Lipschitz norm) of the above set inside the set of all Lipschitz continuous functions.