In the realm of the convergence spaces, the generalisation of topological groups is the convergence groups, and the corresponding extension of the Pontryagin duality is the continuous duality. We prove that local quasi-convexity is a necessary condition for a convergence group to be creflexive. Further, we prove that every character group of a convergence group is locally quasi-convex. 2010 MSC: 43A40; 54A20; 54H11.
{"title":"Duality of locally quasi-convex convergence groups","authors":"Pranav Sharma","doi":"10.4995/AGT.2021.14585","DOIUrl":"https://doi.org/10.4995/AGT.2021.14585","url":null,"abstract":"In the realm of the convergence spaces, the generalisation of topological groups is the convergence groups, and the corresponding extension of the Pontryagin duality is the continuous duality. We prove that local quasi-convexity is a necessary condition for a convergence group to be creflexive. Further, we prove that every character group of a convergence group is locally quasi-convex. 2010 MSC: 43A40; 54A20; 54H11.","PeriodicalId":8046,"journal":{"name":"Applied general topology","volume":"31 1","pages":"193"},"PeriodicalIF":0.8,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87405642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Universal elements are one of the most essential parts in research fields, investigating if there exist (or not) universal elements in different classes of objects. For example, classes of spaces and frames have been studied under the prism of this universality property. In this paper, studying classes of sheaves of Abelian groups, we construct proper universal elements for these classes, giving a positive answer to the existence of such elements in these classes. 2010 MSC: 14F05; 18F20; 54B40.
{"title":"On sheaves of Abelian groups and universality","authors":"S. Iliadis, Y. Sadovnichy","doi":"10.4995/AGT.2021.14422","DOIUrl":"https://doi.org/10.4995/AGT.2021.14422","url":null,"abstract":"Universal elements are one of the most essential parts in research fields, investigating if there exist (or not) universal elements in different classes of objects. For example, classes of spaces and frames have been studied under the prism of this universality property. In this paper, studying classes of sheaves of Abelian groups, we construct proper universal elements for these classes, giving a positive answer to the existence of such elements in these classes. 2010 MSC: 14F05; 18F20; 54B40.","PeriodicalId":8046,"journal":{"name":"Applied general topology","volume":"32 1","pages":"149"},"PeriodicalIF":0.8,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89726386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
When working with a metric space, we are dealing with the additive group (R,+). Replacing (R,+) with an Abelian group (G, ∗), offers a new structure of a metric space. We call it a G-metric space and the induced topology is called the G-metric topology. In this paper, we are studying G-metric spaces based on L-groups (i.e., partially ordered groups which are lattices). Some results in G-metric spaces are obtained. The G-metric topology is defined which is further studied for its topological properties. We prove that if G is a densely ordered group or an infinite cyclic group, then every G-metric space is Hausdorff. It is shown that if G is a Dedekind-complete densely ordered group, (X, d) a G-metric space, A ⊆ X and d is bounded, then f : X → G with f(x) = d(x,A) := inf{d(x, a) : a ∈ A} is continuous and further x ∈ clXA if and only if f(x) = e (the identity element in G). Moreover, we show that if G is a densely ordered group and further a closed subset of R, K(X) is the family of nonempty compact subsets of X, e < g ∈ G and d is bounded, then d′(A,B) < g if and only if A ⊆ Nd(B, g) and B ⊆ Nd(A, g), where Nd(A, g) = {x ∈ X : d(x,A) < g}, dB(A) = sup{d(a,B) : a ∈ A} and d′(A,B) = sup{dA(B), dB(A)}. 2010 MSC: 54C40; 06F20; 16H20.
当处理度量空间时，我们处理的是加性群(R，+)用阿贝尔群(G， *)代替(R，+)，给出了度量空间的一种新结构。我们称它为g度空间诱导的拓扑称为g度拓扑。在本文中，我们研究了基于l群(即偏序群，它们是格)的g度量空间。在g -度量空间中得到了一些结果。定义了G-metric拓扑，并对其拓扑性质进行了进一步研究。证明了如果G是密序群或无限循环群，则每一个G-度量空间都是豪斯多夫空间。证明了如果G是dedekind -完备密序群，(X, d)是G-度量空间，a对X、d有界，则f: X→G，且f(X) = d(X, a):= inf{d(X, a):∈}是连续和进一步x∈clXA当且仅当f (x) = e (G)的单位元素。此外,我们表明,如果G是一个人口进一步命令组和R的一个封闭的子集,K (x)的家庭非空的紧凑的子集(x) e < G∈G和d是有界的,那么d ' (a, B) < G当且仅当⊆Nd (B, G)和B⊆Nd (a、G), Nd (a、G) ={∈x: d (x) < G}, dB (a) =一口{d (a, B):一个∈}和d ' (a, B) =一口{dA (B), dB (a)}。2010 msc: 54c40;06 f20;16净水。
{"title":"Metric spaces related to Abelian groups","authors":"A. Veisi, A. Delbaznasab","doi":"10.4995/AGT.2021.14446","DOIUrl":"https://doi.org/10.4995/AGT.2021.14446","url":null,"abstract":"When working with a metric space, we are dealing with the additive group (R,+). Replacing (R,+) with an Abelian group (G, ∗), offers a new structure of a metric space. We call it a G-metric space and the induced topology is called the G-metric topology. In this paper, we are studying G-metric spaces based on L-groups (i.e., partially ordered groups which are lattices). Some results in G-metric spaces are obtained. The G-metric topology is defined which is further studied for its topological properties. We prove that if G is a densely ordered group or an infinite cyclic group, then every G-metric space is Hausdorff. It is shown that if G is a Dedekind-complete densely ordered group, (X, d) a G-metric space, A ⊆ X and d is bounded, then f : X → G with f(x) = d(x,A) := inf{d(x, a) : a ∈ A} is continuous and further x ∈ clXA if and only if f(x) = e (the identity element in G). Moreover, we show that if G is a densely ordered group and further a closed subset of R, K(X) is the family of nonempty compact subsets of X, e < g ∈ G and d is bounded, then d′(A,B) < g if and only if A ⊆ Nd(B, g) and B ⊆ Nd(A, g), where Nd(A, g) = {x ∈ X : d(x,A) < g}, dB(A) = sup{d(a,B) : a ∈ A} and d′(A,B) = sup{dA(B), dB(A)}. 2010 MSC: 54C40; 06F20; 16H20.","PeriodicalId":8046,"journal":{"name":"Applied general topology","volume":"32 1","pages":"169"},"PeriodicalIF":0.8,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87285264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let X be an arbitrary topological space. F (X) denotes the set of all real-valued functions on X and C(X)F denotes the set of all f ∈ F (X) such that f is discontinuous at most on a finite set. It is proved that if r is a positive real number, then for any f ∈ C(X)F which is not a unit of C(X)F there exists g ∈ C(X)F such that g 6= 1 and f = g f . We show that every member of C(X)F is continuous on a dense open subset of X if and only if every non-isolated point of X is nowhere dense. It is shown that C(X)F is an Artinian ring if and only if the space X is finite. We also provide examples to illustrate the results presented herein. 2010 MSC: 54C40; 13C99.
{"title":"Remarks on the rings of functions which have a finite numb er of di scontinuities","authors":"M. A. Ahmadi Zand, Zahra Khosravi","doi":"10.4995/AGT.2021.14332","DOIUrl":"https://doi.org/10.4995/AGT.2021.14332","url":null,"abstract":"Let X be an arbitrary topological space. F (X) denotes the set of all real-valued functions on X and C(X)F denotes the set of all f ∈ F (X) such that f is discontinuous at most on a finite set. It is proved that if r is a positive real number, then for any f ∈ C(X)F which is not a unit of C(X)F there exists g ∈ C(X)F such that g 6= 1 and f = g f . We show that every member of C(X)F is continuous on a dense open subset of X if and only if every non-isolated point of X is nowhere dense. It is shown that C(X)F is an Artinian ring if and only if the space X is finite. We also provide examples to illustrate the results presented herein. 2010 MSC: 54C40; 13C99.","PeriodicalId":8046,"journal":{"name":"Applied general topology","volume":"5 1","pages":"139"},"PeriodicalIF":0.8,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89594920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we introduce the concept of a soft quasi-pseudometric space. We show that every soft quasi-pseudometric induces a compatible quasi-pseudometric on the collection of all soft points of the absolute soft set whenever the parameter set is finite. We then introduce the concept of soft Isbell convexity and show that a self non-expansive map of a soft quasi-metric space has a nonempty soft Isbell convex fixed point set. 2010 MSC: 03E72; 08A72; 47H10; 54E35; 54E15.
{"title":"On soft quasi-pseudometric spaces","authors":"Hope Sabao, O. O. Otafudu","doi":"10.4995/AGT.2021.13084","DOIUrl":"https://doi.org/10.4995/AGT.2021.13084","url":null,"abstract":"In this article, we introduce the concept of a soft quasi-pseudometric space. We show that every soft quasi-pseudometric induces a compatible quasi-pseudometric on the collection of all soft points of the absolute soft set whenever the parameter set is finite. We then introduce the concept of soft Isbell convexity and show that a self non-expansive map of a soft quasi-metric space has a nonempty soft Isbell convex fixed point set. 2010 MSC: 03E72; 08A72; 47H10; 54E35; 54E15.","PeriodicalId":8046,"journal":{"name":"Applied general topology","volume":"81 1","pages":"17"},"PeriodicalIF":0.8,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77661535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Although fixed point theorems in modular spaces have remarkably applied to a wide variety of mathematical problems, these theorems strongly depend on some assumptions which often do not hold in practice or can lead to their reformulations as particular problems in normed vector spaces. A recent trend of research has been dedicated to studying the fundamentals of fixed point theorems and relaxing their assumptions with the ambition of pushing the boundaries of fixed point theory in modular spaces further. In this paper, we focus on convexity and boundedness of modulars in fixed point results taken from the literature for contractive correspondence and single-valued mappings. To relax these two assumptions, we seek to identify the ties between modular and b-metric spaces. Afterwards we present an application to a particular form of integral inclusions to support our generalized version of Nadler’s theorem in modular spaces. 2010 MSC: 46E30; 47H10; 54C60.
{"title":"Convexity and boundedness relaxation for fixed point theorems in modular spaces","authors":"Fatemeh Lael, S. Shabanian","doi":"10.4995/AGT.2021.13902","DOIUrl":"https://doi.org/10.4995/AGT.2021.13902","url":null,"abstract":"Although fixed point theorems in modular spaces have remarkably applied to a wide variety of mathematical problems, these theorems strongly depend on some assumptions which often do not hold in practice or can lead to their reformulations as particular problems in normed vector spaces. A recent trend of research has been dedicated to studying the fundamentals of fixed point theorems and relaxing their assumptions with the ambition of pushing the boundaries of fixed point theory in modular spaces further. In this paper, we focus on convexity and boundedness of modulars in fixed point results taken from the literature for contractive correspondence and single-valued mappings. To relax these two assumptions, we seek to identify the ties between modular and b-metric spaces. Afterwards we present an application to a particular form of integral inclusions to support our generalized version of Nadler’s theorem in modular spaces. 2010 MSC: 46E30; 47H10; 54C60.","PeriodicalId":8046,"journal":{"name":"Applied general topology","volume":"57 1","pages":"91"},"PeriodicalIF":0.8,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91226733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we define digital homotopic distance and give its relation with LS category of a digital function and of a digital image. Moreover, we introduce some properties of digital homotopic distance such as being digitally homotopy invariance. 2010 MSC: 55M30; 68U10.
{"title":"Digital homotopic distance between digital functions","authors":"A. Borat","doi":"10.4995/AGT.2021.14542","DOIUrl":"https://doi.org/10.4995/AGT.2021.14542","url":null,"abstract":"In this paper, we define digital homotopic distance and give its relation with LS category of a digital function and of a digital image. Moreover, we introduce some properties of digital homotopic distance such as being digitally homotopy invariance. 2010 MSC: 55M30; 68U10.","PeriodicalId":8046,"journal":{"name":"Applied general topology","volume":"80 1","pages":"183"},"PeriodicalIF":0.8,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84116744","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let C∞ (X) denote the family of real-valued continuous functions which vanish at infinity in the sense that {x ∈ X : |f(x)| ≥ 1/n} is compact in X for all n ∈ N. It is not in general true that C∞ (X) is an ideal of C(X). We define those spaces X to be ideal space where C∞ (X) is an ideal of C(X). We have proved that nearly pseudocompact spaces are ideal spaces. For the converse, we introduced a property called “RCC” property and showed that an ideal space X is nearly pseudocompact if and only if X satisfies ”RCC” property. We further discussed some topological properties of ideal spaces.
{"title":"Ideal spaces","authors":"B. Mitra, Debojyoti Chowdhury","doi":"10.4995/agt.2021.13608","DOIUrl":"https://doi.org/10.4995/agt.2021.13608","url":null,"abstract":"<p>Let C<sub>∞ </sub>(X) denote the family of real-valued continuous functions which vanish at infinity in the sense that {x ∈ X : |f(x)| ≥ 1/n} is compact in X for all n ∈ N. It is not in general true that C<span style=\"vertical-align: sub;\">∞ </span>(X) is an ideal of C(X). We define those spaces X to be ideal space where C<span style=\"vertical-align: sub;\">∞ </span>(X) is an ideal of C(X). We have proved that nearly pseudocompact spaces are ideal spaces. For the converse, we introduced a property called “RCC” property and showed that an ideal space X is nearly pseudocompact if and only if X satisfies ”RCC” property. We further discussed some topological properties of ideal spaces.</p>","PeriodicalId":8046,"journal":{"name":"Applied general topology","volume":"17 1","pages":"79"},"PeriodicalIF":0.8,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75555150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article introduces free group representations of planar vortexes in a CW space that are a natural outcome of results for amenable groups and fixed points found by M.M. Day during the 1960s and a fundamental result for fixed points given by L.E.J. Brouwer.
{"title":"Fixed point property of amenable planar vortexes","authors":"J. Peters, T. Vergili","doi":"10.4995/agt.2021.15096","DOIUrl":"https://doi.org/10.4995/agt.2021.15096","url":null,"abstract":"This article introduces free group representations of planar vortexes in a CW space that are a natural outcome of results for amenable groups and fixed points found by M.M. Day during the 1960s and a fundamental result for fixed points given by L.E.J. Brouwer.","PeriodicalId":8046,"journal":{"name":"Applied general topology","volume":"150 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74738559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the class of simple dynamical systems on R induced by continuous maps having finitely many non-ordinary points. We characterize this class using labeled digraphs and dynamically independent sets. In fact, we classify dynamical systems up to their number of non-ordinary points. In particular, we discuss about the class of continuous maps having unique non-ordinary point, and the class of continuous maps having exactly two non-ordinary points.
{"title":"The class of simple dynamics systems","authors":"K. Ali Akbar","doi":"10.4995/AGT.2020.12929","DOIUrl":"https://doi.org/10.4995/AGT.2020.12929","url":null,"abstract":"In this paper, we study the class of simple dynamical systems on R induced by continuous maps having finitely many non-ordinary points. We characterize this class using labeled digraphs and dynamically independent sets. In fact, we classify dynamical systems up to their number of non-ordinary points. In particular, we discuss about the class of continuous maps having unique non-ordinary point, and the class of continuous maps having exactly two non-ordinary points.","PeriodicalId":8046,"journal":{"name":"Applied general topology","volume":"44 1","pages":"215-233"},"PeriodicalIF":0.8,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80714008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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