Recent zbMATH articles in MSC 54Ahttps://zbmath.org/atom/cc/54A2021-06-15T18:09:00+00:00WerkzeugOn \(\psi_\mathcal{H} (.)\)-operator in weak structure spaces with hereditary classes.https://zbmath.org/1460.540032021-06-15T18:09:00+00:00"Abu-Donia, Hassan Mustafa"https://zbmath.org/authors/?q=ai:abu-donia.hassan-mustafa"Hosny, Rodyna A."https://zbmath.org/authors/?q=ai:hosny.rodyna-aSummary: Weak structure space (briefly, \(wss\)) has master looks, when the whole space is not open, and these classes of subsets are not closed under arbitrary unions and finite intersections, which classify it from typical topology. Our main target of this article is to introduce \(\psi_\mathcal{H}(.)\)-operator in hereditary class weak structure space (briefly, \(\mathcal{H} wss) (X, w, \mathcal{H})\) and examine a number of its characteristics. Additionally, we clarify some relations that are credible in topological spaces but cannot be realized in generalized ones. As a generalization of \(w\)-open sets and \(w\)-semiopen sets, certain new kind of sets in a weak structure space via \(\psi_\mathcal{H}(.)\)-operator called \(\psi_\mathcal{H}\)-semiopen sets are introduced. We prove that the family of \(\psi_\mathcal{H}\)-semiopen sets composes a supra-topology on \(X\). In view of hereditary class \(\mathcal{H}_0, w T_1\)-axiom is formulated and also some of their features are investigated.\(L\)-fuzzy pre-proximities, \(L\)-fuzzy filters and \(L\)-fuzzy grills.https://zbmath.org/1460.540052021-06-15T18:09:00+00:00"Ramadan, Ahmed Abd El-Kader"https://zbmath.org/authors/?q=ai:ramadan.ahmed-abd-el-kader"Usama, M. A."https://zbmath.org/authors/?q=ai:usama.m-a"Abd El-Latif, Ahmed Aref"https://zbmath.org/authors/?q=ai:abd-el-latif.ahmed-arefSummary: This article gives results on fixed complete lattice \(L\)-fuzzy pre-proximities, \(L\)-fuzzy grills and \(L\)-fuzzy filters. Moreover, we investigate the relations among the \(L\)-fuzzy pre-proximities, \(L\)-fuzzy grills and \(L\)-fuzzy filters. We show that there is a Galois correspondence between the category of separated \(L\)-fuzzy grill spaces and that of separated \(L\)-fuzzy pre-proximity spaces. We introduced the local function associated with \(L\)-fuzzy grill and \(L\)-fuzzy topology and studied some of its properties. Finally, we build an \(L\)-fuzzy topology for the corresponding \(L\)-fuzzy grill by using local function.Fuzzy fixed point theorems in ordered cone metric spaces.https://zbmath.org/1460.540352021-06-15T18:09:00+00:00"Azam, Akbar"https://zbmath.org/authors/?q=ai:azam.akbar"Mehmood, Nayyar"https://zbmath.org/authors/?q=ai:mehmood.nayyar"Rashid, Maliha"https://zbmath.org/authors/?q=ai:rashid.maliha"Radenović, Stojan"https://zbmath.org/authors/?q=ai:radenovic.stojanSummary: In this article, we introduce the notion of the multivalued fuzzy mappings satisfying w.l.b property and l.b properties and prove some results for multivalued generalized contractive fuzzy mappings in ordered-cone metric spaces without the assumption of normality on cones. We generalize many results in the literature.On certain versions of straightness.https://zbmath.org/1460.540182021-06-15T18:09:00+00:00"Das, Pratulananda"https://zbmath.org/authors/?q=ai:das.pratulananda"Pal, Sudip Kumar"https://zbmath.org/authors/?q=ai:pal.sudip-kumar"Adhikary, Nayan"https://zbmath.org/authors/?q=ai:adhikary.nayanAll spaces in the sequel are metric and \(C(X)\) denotes the set of all continuous real-valued functions on \(X\). \textit{A. Berarducci} et al. [Topology Appl. 146--147, 339--352 (2005; Zbl 1078.54014)] introduced and studied the notion of straight spaces: A space \(X\) is \textit{straight} if whenever \(X\) is the union of two closed sets, then \(f\in C(X)\) is uniformly continuous iff its restriction to each of the closed sets is uniformly continuous.
In this paper, the authors consider notions of continuity which are strictly weaker than the notion of uniform continuity, and introduce three versions of straightness, namely pre-straightness, pre\((\ast)\)-straightness and \(W\)-straightness: A space \(X\) is \textit{pre-straight} if whenever \(X\) is the union of closed sets, then \(f\in C(X)\) is Cauchy regular iff its restriction to each of the closed sets is Cauchy regular, where a mapping \(f:X\to Y\) is Cauchy regular if for any Cauchy sequence \(\{x_n\}\) in \(X\), \(\{f(x_n)\}\) is Cauchy in \(Y\). A space \(X\) is \textit{pre\((\ast)\)-straight} if whenever \(X\) is the union of closed sets, then a real-valued Cauchy regular function \(f\) is uniformly continuous iff its restriction to each of the closed sets is uniformly continuous. Recall that a sequence \(\{x_n\}\) in a metric space \((X, d)\) is \textit{quasi-Cauchy} if given \(\varepsilon>0\), there is a natural number \(n_0\) such that \(d(x_{n+1},x_n)<\varepsilon\) for all \(n\geq n_0\). A space \(X\) is \textit{\(W\)-straight} if whenever \(X\) is the union of closed sets, then \(f\in C(X)\) is ward continuous iff its restriction to each closed sets is ward continuous, where a mapping \(f: X\to Y\) is \textit{ward continuous} if it preserves quasi-Cauchy sequences.
The authors investigate some properties of these spaces and the relationship between these notions. For instance, they obtain the following: A space \(X\) is straight if and only if \(X\) is pre-straight and pre\((\ast)\)-straight. A space \(X\) is pre\((\ast)\)-straight if and only if its completion \(\widehat{X}\) is straight.
Reviewer: Kohzo Yamada (Shizuoka)Generalized \(w\) closed sets in biweak structure spaces.https://zbmath.org/1460.540022021-06-15T18:09:00+00:00"Abu-Donia, Hassan Mustafa"https://zbmath.org/authors/?q=ai:abu-donia.hassan-mustafa"Hosny, Rodyna A."https://zbmath.org/authors/?q=ai:hosny.rodyna-aSummary: As a generalization of the classes of \(gw\) closed (resp. \(gw\) open, \(sgw\) closed) sets in a weak structure space \((X, w)\), the notions of \(ij\)-generalized \(w\) closed (resp. \(ij\)-generalized \(w\) open, \(ij\)-strongly generalized \(w\) closed) sets in a biweak structure space \((X, w_1, w_2)\) are introduced. In terms of these concepts, new forms of continuous functions between biweak spaces are constructed. Additionally, the concepts of \(ij\)-\(w\) normal, \(ij\)-\(gw\) normal, \(ij\)-\(wT_{\frac{1}{2}}\), and \(ij\)-\(w^{\sigma} T_{\frac{1}{2}}\) spaces are studied and several characterizations of them are acquired.Observability in locally vague environments.https://zbmath.org/1460.681092021-06-15T18:09:00+00:00"Demirci, Mustafa"https://zbmath.org/authors/?q=ai:demirci.mustafaSummary: In this paper, we introduce observable L-fuzzy subsets of an L-fuzzy set in a locally vague environment, and give their axiomatization. In addition to this, lower and upper observability operators that enable us to approximate non-observable L-fuzzy sets within some observable bounds are studied. In particular, we deal with their topological characterization, and expose that they can be identified with the stratified L-interior and the stratified L-closure operators of a stratified Alexandroff L-topology on an L-fuzzy set.On \(\mathcal{I}\)-quotient mappings and \(\mathcal{I}\)-\(cs'\)-networks under a maximal ideal.https://zbmath.org/1460.540152021-06-15T18:09:00+00:00"Zhou, Xiangeng"https://zbmath.org/authors/?q=ai:zhou.xiangengIn the present paper, for an ideal \(I\) on \(\mathbb{N}\) and a mapping \(f:X\rightarrow Y\), the notions of \(I\)-quotient mapping and \(I\)-\(cs' \)-network for a topological space are introduced. Also, properties of the notions of \(I\)-quotient mappings and \(I\)-\(cs' \)-networks are studied.
Reviewer: Erdal Ekici (Çanakkale)\(\psi^*\)-closed sets in fuzzy topological spaces.https://zbmath.org/1460.540042021-06-15T18:09:00+00:00"Allah, M. A. Abd"https://zbmath.org/authors/?q=ai:allah.m-a-abd"Nawar, A. S."https://zbmath.org/authors/?q=ai:nawar.ashraf-sSummary: In this paper, we introduce a new class of fuzzy sets, namely, fuzzy \(\psi^*\)-closed sets for fuzzy topological spaces, and some of their properties have been proved. Further, we introduce fuzzy \(\psi^*\)-continuous, fuzzy \(\psi^*\)-irresolute functions, and fuzzy \(\psi^*\)-closed (open) functions, as applications of these fuzzy sets, fuzzy \(T_{1/5}\)-spaces, fuzzy \({T}_{1/5}^{\psi \ast}\)-spaces, and fuzzy \(^{\psi *} T_{1/5}\)-spaces.{\(\alpha\)}-type fuzzy \(H\)-contractive mappings in fuzzy metric spaces.https://zbmath.org/1460.540372021-06-15T18:09:00+00:00"Beg, I."https://zbmath.org/authors/?q=ai:beg.ismat"Gopal, D."https://zbmath.org/authors/?q=ai:gopal.dhananjay"Došenović, T."https://zbmath.org/authors/?q=ai:dosenovic.tatjana"Rakić, D."https://zbmath.org/authors/?q=ai:rakic.dusanSummary: We introduce a new concept of {\(\alpha\)}-fuzzy \(\mathcal{H}\)-contractive mapping which is essentially weaker than the class of fuzzy contractive mapping and stronger than the concept of {\(\alpha\)}-{\(\phi\)}-fuzzy contractive mapping. For this type of contractions, the existence and uniqueness of fixed point in fuzzy \(M\)-complete metric spaces is also established.Star versions of the Rothberger property on hyperspaces.https://zbmath.org/1460.540072021-06-15T18:09:00+00:00"Casas-de la Rosa, Javier"https://zbmath.org/authors/?q=ai:casas-de-la-rosa.javier"Martínez-Ruiz, Iván"https://zbmath.org/authors/?q=ai:martinez-ruiz.ivan"Ramírez-Páramo, Alejandro"https://zbmath.org/authors/?q=ai:ramirez-paramo.alejandroA space \(X\) is said to be star-Rothberger (resp., strongly star-Rothberger) if for each sequence \((\mathcal U_n)\) of open covers of \(X\) there is a sequence \((U_n)\) (resp., \((x_n)\)) such that for each \(n\), \(U_n \in \mathcal U_n\) (resp., \(x_n \in X\)) and \(\{St(U_n,\mathcal U_n): n \in \mathbb N\}\) (resp., \(\{St(x_n,\mathcal U_n): n \in \mathbb N\}\)) is an open cover of \(X\). The authors characterize the star-Rothberger and strongly star-Rothberger properties of the hyperspace over a space \(X\) equipped with the Fell topology in terms of selective properties of \(X\) of \(\pi\)-network-type. They also characterize the strongly star-Rothberger property on hyperspaces endowed with the lower Vietoris topology. Selective versions of metacompactness and and mesocompactness are also considered.
Reviewer: Ljubiša D. Kočinac (Niš)Lecture notes on general topology.https://zbmath.org/1460.540012021-06-15T18:09:00+00:00"Wang, Guoliang"https://zbmath.org/authors/?q=ai:wang.guoliangPublisher's description: This book is intended as a one-semester course in general topology, a.k.a. point-set topology, for undergraduate students as well as first-year graduate students. Such a course is considered a prerequisite for further studying analysis, geometry, manifolds, and certainly, for a career of mathematical research. Researchers may find it helpful especially from the comprehensive indices.
General topology resembles a language in modern mathematics. Because of this, the book is with a concentration on basic concepts in general topology, and the presentation is of a brief style, both concise and precise. Though it is hard to determine exactly which concepts therein are basic and which are not, the author makes efforts in the selection according to personal experience on the occurrence frequency of notions in advanced mathematics, and to related books that have received admirable reviews.
This book also contains exercises for each chapter with selected solutions. Interrelationships among concepts are taken into account frequently. Twelve particular topological spaces are repeatedly exploited, which serve as examples to learn new concepts based on old ones.Rothberger and Rothberger-type star selection principles on hyperspaces.https://zbmath.org/1460.540082021-06-15T18:09:00+00:00"Díaz-Reyes, Jesús"https://zbmath.org/authors/?q=ai:reyes.jesus-diaz"Ramírez-Páramo, Alejandro"https://zbmath.org/authors/?q=ai:ramirez-paramo.alejandro"Tenorio, Jesús F."https://zbmath.org/authors/?q=ai:tenorio.jesus-fFor a topological space \(X\), let \(CL (X)\) denote the hyperspace consisting of all nonempty closed subsets of \(X\) endowed with the Vietoris topology, and let \(\mathbb{K}(X)\) (resp., \(\mathbb{F}(X)\), \(\mathbb{CS}(X)\)) be the subspace consisting of all nonempty compact subsets (resp., nonempty finite subsets, convergent sequences) of \(X\). Motivated by the work of \textit{Z. Li} [Topology Appl. 212, 90--104 (2016; Zbl 1355.54014)], the authors characterize the Rothberger property and two selection principles called star-Rothberger and strongly star-Rothberger in the spaces \(CL (X)\), \(\mathbb{K}(X)\), \(\mathbb{F}(X)\) and \(\mathbb{CS}(X)\).
Let \(X\) be a topological space and let \(\Lambda\) be one of the hyperspaces \(CL (X)\), \(\mathbb{K}(X)\), \(\mathbb{F}(X)\) and \(\mathbb{CS}(X)\). Let \(\mathbf{S}_1(\mathcal{A},\mathcal{B})\) be the selection principle defined by \textit{M. Scheepers} [Topology Appl. 69, No. 1, 31--62 (1996; Zbl 0848.54018)]. The authors introduce the notion of \(\pi_V(\Lambda)\)-network modifying the notion of \(\pi_V\)-network defined by Li [loc. cit.], and two selection principles \(\mathbf{S}_{\Pi_V} (\Pi_V(\Lambda),\Pi_V(\Lambda))\) and \(\mathbf{S}^*_{\Pi_V} (\Pi_V(\Lambda),\Pi_V(\Lambda))\), where \(\Pi_V (\Lambda)\) is the collection of \(\pi_V(\Lambda)\)-networks of \(X\). The following theorems are proved: \(\Lambda\) has the Rothberger property if and only if \(X\) satisfies \(\mathbf{S}_{1} (\Pi_V(\Lambda),\Pi_V(\Lambda))\); \(\Lambda\) is strongly star-Rothberger if and only if \(X\) satisfies \(\mathbf{S}_{\Pi_V} (\Pi_V(\Lambda),\Pi_V(\Lambda))\); and \(\Lambda\) is star-Rothberger if and only if \(X\) satisfies \(\mathbf{S}^*_{\Pi_V} (\Pi_V(\Lambda),\Pi_V(\Lambda))\). Let \(\mathcal{D}\) denote the family of dense subsets of a given space. The authors also give a characterization of the selection principle \(\mathbf{S}_1 (\mathcal{D},\mathcal{D})\) for \(\Lambda\) following ideas by Li [loc. cit.].
Reviewer: Takamitsu Yamauchi (Matsuyama)