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If f is a continuous selection for the Vietoris hyperspace F (X) of the nonempty closed subsets of a space X, then the point f (X) ∈ X is not as arbitrary as it might seem at first glance. In this paper, we will characterise these points by local properties at them. Briefly, we will show that p = f (X) is a strong butterfly point precisely when it has a countable clopen base in ⊂ for some open set U ⊂ X \ {p} with ⊂ = U ⊂ {p}. Moreover, the same is valid when X is totally disconnected at p = f (X) and p is only assumed to be a butterfly point. This gives the complete affirmative solution to a question raised previously by the author. Finally, when p = f (X) lacks the above local base-like property, we will show that F(X) has a continuous selection h with the stronger property that h(S) = p for every closed S ⊂ X with p ∈ S. [ABSTRACT FROM AUTHOR]

The paper deals with some further results concerning the class of two-Lipschitz operators. We prove first an isometric isomorphism identification of two-Lipschitz operators and Lipschitz operators. After defining and characterizing the adjoint of a two-Lipschitz operator, we prove a Schauder type theorem on the compactness of the adjoint. We study the extension of two-Lipschitz operators from the cartesian product of two complemented subspaces of a Banach space to the cartesian product of whole spaces. Also, we show that every two-Lipschitz functional defined on the cartesian product of two pointed metric spaces admits an extension with the same two-Lipschitz norm under some requirements on domaine spaces. [ABSTRACT FROM AUTHOR]

This paper continues a series in which we study deficiencies in previously published works concerning fixed point assertions for digital images. [ABSTRACT FROM AUTHOR]

Fixed point theorems are very important tools in different branches of mathematics. In this paper, we introduce partial uniform spaces as a generalization of metric spaces; and study some basic properties. Various examples support the theory. We prove fixed point theorems for H-partial uniform spaces by using a map called an E-distance function. Finally, we give the applications of these fixed point theorems to compress digital images. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the class of large MR-Kannan contractions on Banach space that contains the classes of Kannan, enriched Kannan, large Kannan, MR-Kannan contractions and some other classes of nonlinear operators. Some examples are presented to support the concepts introduced herein. We prove the existence of a unique fixed point for such a class of operators in Banach spaces. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study some variants of nowhere dense sublocales called maximal nowhere dense and homogeneous maximal nowhere dense sublocales. These concepts were initially introduced by Veksler in classical topology. We give some general properties of these sublocales and further examine their relationship with both inaccessible sublocales and remote sublocales. It turns out that a locale has all of its non-void nowhere dense sublocales maximal nowhere dense precisely when all of its its non-void nowhere dense sublocales are inaccessible. We show that the Booleanization of a locale is inaccessible with respect to every dense and open sublocale. In connection to remote sublocales, we prove that, if the supplement of an open dense sublocale S is homogeneous maximal nowhere dense, then every S#-remote sublocale is *-remote from S. Every open localic map that sends dense elements to dense elements preserves and reflects maximal nowhere dense sublocales. If such a localic map is further injective, then it sends homogeneous maximal nowhere dense sublocales back and forth. [ABSTRACT FROM AUTHOR]

Given a metric continuum X, we consider the collection of all regular subcontinua of X and the collection of all meager subcontinua of X, these hyperspaces are denoted by D(X) and M(X), respectively. It is known that D(X) is compact if and only if D(X) is finite. In this way, we find some conditions related about the cardinality of D(X) and we reduce the fact to count the elements of D(X) to a Graph Theory problem, as an application of this, we prove in particular that |D(X)| G {2, 3,4, 5, 8, 9} for any continuum X. Also, we prove that D(X) is never homeomorphic to N. On the other hand, given a point p G X, we consider the meager composant and the filament composant of p in X, denoted by Mpf and FcsX (p), respectively, and we study some relations between Mpx and FcsX (p) such as the equality of them as a subset of X. Also, we construct examples showing that the collection Fcs(X) = {FcsX (p): p G X} can be homeomorphic to: any finite discrete space, the harmonic sequence, the closure of the harmonic sequence and the Cantor set. Finally, we study the contractibility of M(X); we prove the arc of pseudo-arcs, which is a no contractible continuum, satisfies that its hyperspace of meager subcontinua is contractible, given a solution to Problem 3 of [10]. Most of the results shown in this paper are focus to answer problems and questions posed in [6], [9] and [10]. Also, we rise open problems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the topological properties of algebraic sets with zero divisors. We impose a subbasic topology on the set of proper ideals of a k-algebra and this new "k-space" becomes a generalization of the corresponding Zariski space. We prove that a k-space is T0, quasi-compact, spectral, and connected. Moreover, we study continuous maps between such k-spaces. We conclude with a question about construction of a sheaf of k-spaces similar to affine schemes. [ABSTRACT FROM AUTHOR]

In this paper we provide some general results about topological spaces X satisfying any of the following properties: starn-P, weakly starn-P or almost starn-P, for P C {kCC,WkL, DkCC}, where k is an infinite cardinal number. The particular cases when k = u, P C {ccc, weakly Lindelof, Dccc} are obtained. Furthermore, for the same classes of spaces defined by such P, by applying Erdos-Rado's theorem and using the rank l-diagonal notion, we establish some cardinal inequalities. [ABSTRACT FROM AUTHOR]

In this paper we introduce some weak dynamical properties by using subbases for the phase space. Among them, the notion of light chaos is the most significant. Several examples, which clarify the relationships between this kind of chaos and some classical notions, are given. Particular attention is also devoted to the connections between the dynamical properties of a system and the dynamical properties of the associated functional envelope. We show, among other things, that a continuous map f: X → X, where X is a metric space, is chaotic (in the sense of Devaney) if and only if the associated functional dynamical system is lightly chaotic. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the sufficient conditions for the existence of attractor of a generalized cyclic iterated function system composed of a complete metric space and a finite collection of generalized cyclic Γ-contraction mappings. Some examples are presented to support our main results and concepts defined herein. The results proved in the paper extend and generalize various well known results in the existing literature. [ABSTRACT FROM AUTHOR]

Several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces. We show that in many cases, researchers using these tools have derived conclusions that are incorrect, trivial, or limited. [ABSTRACT FROM AUTHOR]

In the present paper we define the degree of nondensifiability (DND for short) of a bounded linear operator T on a Banach space and analyze its properties and relations with the Hausdorff measure of noncompactness (MNC for short) of T. As an application of our results, we have obtained a formula to find the essential spectral radius of a bounded operator T on a Banach space as well as we have provided the best possible lower bound for the Hyers-Ulam stability constant of T in terms of the aforementioned DND. [ABSTRACT FROM AUTHOR]

Consider the ring M∘(X,μ) of functions which are discontinuous on a set of measure zero which is introduced and studied extensively in [2]. In this paper, we introduce a ring B1(X,μ) of functions which are pointwise limits of sequences of functions in Mo(X,μ'). We study various properties of zero sets, B1(X, p)-separated and B1(X, p)-embedded subsets of B1(X,μ) and also establish an analogous version of Urysohn's extension theorem. We investigate a connection between ideals of B1(X,μ) and ZB-filters on X. We study an analogue of Gelfand-Kolmogoroff theorem in our setting. We define real maximal ideals of B1(X,μ) and establish the result |RMax(Mo(X,μ))| = |RMax(B1(X, μ))|, where RMax(M∘(X, μ)) and RMax(B1(X, μ)) are the sets of all real maximal ideals of M∘(X,μ) and B1(X,μ), respectively. [ABSTRACT FROM AUTHOR]

In this paper we study the space of strongly Lipschitz (lp,lq)-factorable operators between metric spaces and a Banach spaces. In particular, a factorization of this class through lp and lq spaces is given. We show that this type of operators fits in the theory of composition a-Banach Lipschitz operator ideal. As a special case, we get a Lipschitz version of weakly p-nuclear operators. [ABSTRACT FROM AUTHOR]

This paper presents some existence and uniqueness results for a solution of a system of equations. Our results extend and generalize the well-known and celebrated results of Boyd and Wong [Proc. Amer. Math. Soc. 20 (1969)], Matkowski [Dissertations Math. (Rozprawy Mat.) 127 (1975)], Proinov [Nonlinear Anal. 64 (2006)], Ri [Indag. Math. (N. S.) 27 (2016)] and many others. We also present some illustrative examples to validate our results. [ABSTRACT FROM AUTHOR]

Given a uniform space (X, U), we denote by F*(X) to the family of fuzzy sets u in (X, U) such that u is normal and upper semicontinu-ous. Let UE be the endograph uniformity on F*(X). In this paper, we mainly characterize totally bounded and compact subsets in the uniform space (F*(X), UE). [ABSTRACT FROM AUTHOR]

In this paper new notions of dimensions for ideal topological spaces are inserted, called *-quasi covering dimension and ideal quasi covering dimension. We study several of their properties and investigate their relations with types of covering dimensions like the-covering dimension and the ideal covering dimension. [ABSTRACT FROM AUTHOR]

The paper deals with some further results concerning the class of two-Lipschitz operators. We prove first an isometric isomorphism identification of two-Lipschitz operators and Lipschitz operators. After defining and characterizing the adjoint of a two-Lipschitz operator, we prove a Schauder type theorem on the compactness of the adjoint. We study the extension of two-Lipschitz operators from the cartesian product of two complemented subspaces of a Banach space to the cartesian product of whole spaces. Also, we show that every two-Lipschitz functional defined on the cartesian product of two pointed metric spaces admits an extension with the same two-Lipschitz norm under some requirements on domaine spaces. [ABSTRACT FROM AUTHOR]

Recently, Karapinar introduced a new Hardy-Rogers type contractive mapping using the concept of interpolation and proved a fixed point theorem in complete metric space. This new type of mapping, called "interpolative Hardy-Rogers type contractive mapping" is a generalization of Hardy-Rogers's fixed point theorem. Following this direction of research, in this paper, we will present some fixed point results of Hardy-Rogers-type for cyclic mappings on complete metric spaces. Moreover, an example is given to illustrate the usability of the obtained results. [ABSTRACT FROM AUTHOR]

This paper continues a series discussing flaws in published assertions concerning fixed points in digital images. [ABSTRACT FROM AUTHOR]

In this paper, we introduce generalizations of the concept of MR-Kannan type contractions and utilize those conditions to derive new fixed point theorems under both contractive and non-contractive conditions. Our approach enhances various existing results related to enriched mappings. [ABSTRACT FROM AUTHOR]

For a Tychonoff space X, the lattice UX was introduced in L. A. Pérez-Morales, G. Delgadillo-Piñón, and R. Pichardo-Mendoza, The lattice of uniform topologies on C(X), Questions and Answers in General Topology 39 (2021), 65-71. In the present paper we continue the study of UX. To be specific, the present paper deals, in its first half, with structural and categorical properties of UX, while in its second part focuses on cardinal characteristics of the lattice and how these relate to some cardinal functions of the space X. [ABSTRACT FROM AUTHOR]

This paper introduces proximal path cycles, which lead to the main results in this paper, namely, extensions of the Mitsuishi-Yamaguchi Good Coverning Theorem with different forms of Tanaka good cover of an Alexandrov space equipped with a proximity relation as well as extension of the Jordan curve theorem. In this work, a path cycle is a sequence of maps h1,...,hi,...,hn-1 mod n in which hi : [ 0,1 ] → X and hi(1) = hi+1(0) provide the structure of a path-connected cycle that has no end path. An application of these results is also given for the persistence of proximal video frame shapes that appear in path cycles. [ABSTRACT FROM AUTHOR]

Whenever the closure of an open set is also open, it is called e-open and if a space have a base consisting of e-open sets, it is called e-space. In this paper we first introduce and study e-spaces and e-continuous func-tions (we call a function f from a space X to a space Y an e-continuous at x 2 X if for each open set V containing f(x) there is an e-open set containing x with f(U) C V). We observe that the quasicomponent of each point in a space X is determined by e-continuous functions on X and it is characterized as the largest set containing the point on which every e-continuous function on X is constant. Next, we study the rings Ce(X) of all real valued e-continuous functions on a spaCe X. It turns out that Ce(X) coincides with the ring of real valued clopen continuous functions on X which is a C(Y) for a zero-dimensional spaCe Y whose elements are the quasicomponents of X. Using this fact we characterize the real maximal ideals of Ce(X) and also give a natural representation of its maximal ideals. Finally we have shown that Ce(X) determines the topology of X if and only if it is a zero-dimensional spaCe. [ABSTRACT FROM AUTHOR]

In this paper, f-contractions are defined and then, some new fixed point theorems are established for certain nonlinear mappings associated with one-dimensional Φ-comparison functions in fuzzy metric spaces. Next, generalized f-contractions are defined by using five-dimensional Φ-comparison functions, and the existence of fixed points for nonlinear maps on fuzzy metric spaces is studied. Moreover, some examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we prove some fixed point theorems for Lipschitz type mappings in the setting of metric spaces. Our results open up the unex-plored area of fixed points of Lipschitz type mappings for investigation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate shift spaces arising from a multidimensional graph G. In particular, we investigate non-emptiness and existence of periodic points for a multidimensional shift space. We derive sufficient conditions under which these questions can be answered affirmatively. We investigate the structure of the shift space using the generating matrices. We prove that any d-dimensional shift of finite type is finite if and only if it is conjugate to a shift generated through permutation matrices. We prove that if any triangular pattern of the form a cb can be extended to a 1 x 1 square then the two dimensional shift space possesses periodic points. We introduce the notion of an E-pair for a two dimensional shift space. Using the notion of an E-pair, we derive sufficient conditions for non-emptiness of the two dimensional shift space under discussion. [ABSTRACT FROM AUTHOR]

In this paper we consider some recent relative versions of Menger property called set strongly star Menger and set star Menger properties and the corresponding Hurewicz-type properties. In particular, using [2], we easily" prove that the set strong star Menger and set strong star Hurewicz properties are between countable compactness and the property of having countable extent. Also we show that the extent of a regular set star Menger or a set star Hurewicz space cannot exceed c. Moreover, we construct (1) a consistent example of a set star Menger (set star Hurewicz) space which is not set strongly star Menger (set strongly star Hurewicz) and show that (2) the product of a set star Menger (set star Hurewicz) space with a compact space need not be set star Menger (set star Hurewicz). In particular, (1) and (2) answer some questions posed by Kocinac, Konca and Singh in [17] and [23]. [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of some topological and cardinal properties of topological spaces under the influence of the NτΦ-kernel of a space X. It has been proved that the NτΦ-kernel of a space X preserves the density and the network p - weight of normal spaces. Besides, shown that the N-compact kernel of a space X preserves the Souslin properties, the weight, the density, and the p-network weight of normal spaces. [ABSTRACT FROM AUTHOR]

In this paper, the concept of rough representation of a rough topological group on a Banach space is explored. Mainly, the continuity and the irreducibility of rough representations are studied. [ABSTRACT FROM AUTHOR]

In this paper, we introduce four new types of contractions called in this order Kannan-type cyclic contraction via interpolation, interpolative Ćirić-Reich-Rus type cyclic contraction, and we prove the existence and uniqueness for a fixed point for each situation. [ABSTRACT FROM AUTHOR]

This paper continues a series discussing aws in published assertions concerning fixed points in digital metric spaces. [ABSTRACT FROM AUTHOR]

In this present paper, besides other things, we introduce the concept of Ćirić-generalized contractions via wt-distance and then we will prove some new fixed point results for these mappings, which generalize and improve fixed point theorems by L. B. Ćirić in [9, 8, 10] and also, B. E. Rhoades in [23]. Some examples illustrate usefulness of the new results. At the end, we will give some applications to nonlinear fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some concepts of ⋆-(quasi)-pseudometric spaces, and give an example which shows that there is a ⋆-quasipseudometric space which is not a quasi-pseudometric space. We also study the conditions under which ⋆-quasi-pseudometric semitopological groups are paratopological groups or topological groups. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of fixed point results for set-valued mappings in Menger probabilistic metric spaces equipped with an amorphous binary relation and a Hadzic-type t-norm. For the usability of such findings we present a Kelisky-Rivlin type result for a class of Bernstein type special operators introduced by Deo et. al. [Appl. Math. Comput. 201, (2008), 604-612] on the space C([0, n+i]). In this way, these investigations extend, modify and generalize some prominent recent fixed point results of the existing literature. [ABSTRACT FROM AUTHOR]

Several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces and homotopy invariant fixed point theory. We show that some of the published assertions based on these tools are incorrect or trivial; we offer improvements on others. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new iteration process which is faster than the leading; S [J. Nonlinear Convex Anal. 8, no. 1 (2007), 61{79], Thakur et al. [App. Math. Comp. 275 (2016), 147{155] and M [Filomat 32, no. 1 (2018), 187{196] iterations for numerical reckoning fixed points. Using this new iteration process, some fixed point convergence results for generalized ff-nonexpansive mappings in the setting of uniformly convex Banach spaces are proved. At the end of paper, we offer a numerical example to compare the rate of convergence of the proposed iteration process with the leading iteration processes. [ABSTRACT FROM AUTHOR]

In this paper, we define uniformities and topologies as relators and show the equivalences of these definitions with the classical ones. For this, we summarize the essential properties of relators, using their theory from earlier works of Á. Száz. Moreover, we prove implications between important topological properties of relators and disprove others. Finally, we show that our earlier analogous definition [G. Pataki, Investigation of proximal spaces using relators, Axioms 10, no. 3 (2021): 143.] for uniformly and proximally filtered property is equivalent to the topological one. At the end of this paper, uniformities and topologies are defined in the same way. This will give us new possibilities to compare these and other topological structures. [ABSTRACT FROM AUTHOR]

An almost P-space is a topological space in which every zero-set is regular-closed. We introduce a large class of spaces, C-almost P-space (briey CAP-space), consisting of those spaces in which the closure of the interior of every zero-set is a zero-set. In this paper we study CAP-spaces. It is proved that if X is a dense and Z#-embedded sub-space of a space T, then T is CAP if and only if X is a CAP and CRZ-extended in T (i.e, for each regular-closed zero-set Z in X, clT Z is a zero-set in T). In 6P.5 of [8] it was shown that a closed count- able union of zero-sets need not be a zero-set. We call X a CZ-space whenever the closure of any countable union of zero-sets is a zero-set. This class of spaces contains the class of P-spaces, perfectly normal spaces, and is contained in the cozero complemented spaces and CAP-spaces. In this paper we study topological properties of CZ (resp. coz-ero complemented)-space and other classes of topological spaces near to them. Some algebraic and topological equivalent conditions of CZ (resp. cozero complemented)-space are characterized. Examples are provided to illustrate and delimit our results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of fuzzy (F, φ,β-ψ)-contractive mappings in fuzzy metric spaces and utilize the same to prove some existence and uniqueness fuzzy φ-fixed point results in both M-complete and G-complete fuzzy metric spaces. The obtained results extend, generalize and improve some relevant results of the existing literature. An illustrative example is utilized to demonstrate the usefulness and effectiveness of our results. [ABSTRACT FROM AUTHOR]

In this paper, we obtain two common fixed point results for mappings satisfying the generalized (ψ, φ)-contractive type conditions given by a rational expression on a complete metric space. Our results generalize several well known theorems of the literature in the context of (ψ, φ)- rational contraction. In addition, there is an example for obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate new solutions to the Rhoades' discontinuity problem on the existence of a self-mapping which has a fixed point but is not continuous at the fixed point on metric spaces. To do this, we use the number defined as n(x,y)=[d(x,y)]β[d(x,Ty)]α[d(x,Ty)]γ[(d(x,Ty)+d(x,Ty))/2]1−α−β−γ, where α, β, γ ∈ ( 0,1 ) with α + β + γ < 1 and some interpolative type contractive conditions. Also, we investigate some geometric properties of Fix(T) under some interpolative type contractions and prove some fixed-disc (resp. fixed-circle) results. Finally, we present a new application to the discontinuous activation functions. [ABSTRACT FROM AUTHOR]

In this paper we introduce notions of digital semicovering and digital quasicovering maps. We show that these are generalizations of digital covering maps and investigate their relations. We will also clarify the relationship between these generalizations and digital path lifting. [ABSTRACT FROM AUTHOR]

In this paper we continue to research relationships between closure-type properties of hyperspaces over a space X and covering properties of X. For a Hausdorff space X we denote by 2X the family of all closed subsets of X. We investigate selection properties of the bitopological space (2X, Δ1+, Δ2+) where Δi+ is the upper Δi-topology for each i=1,2. [ABSTRACT FROM AUTHOR]

One of the fundamental problems in rings of continuous functions is to extract those spaces X for which C(X) determines X, that is to investigate X and Y such that C(X) isomorphic with C(Y ) implies X homeomorphic with Y . Later S. Banach and M. Stone proved independently with slight variance, that if X is a compact Hausdorff space, C(X) also determine X. Their works were maximally extended by E. Hewitt who introduced realcompact spaces and later Melvin Henriksen and Biswajit Mitra solved the problem for locally compact and nearly realcompact spaces. In this paper we tried to develop a unified theory of this problem to cover up all the works in the literature introducing the notion P-compact spaces. [ABSTRACT FROM AUTHOR]

In this paper the network-type properties (network, cs−network, cs∗−network, cn−network and ck−network) of the space SPn GX of Gpermutation degree of X are studied. It is proved that: (1) If X is a T1-space that has a network of cardinality ≤ κ, then SPnGX has a network of cardinality ≤ κ; (2) If X is a T1-space that has a cs-network (resp. cs∗ -network) of cardinality ≤ κ, then SPn GX has a cs-network (resp. cs∗ -network) of cardinality ≤ κ; (3) If X is a T1-space that has a cn-network (resp. ck-network) of cardinality ≤ κ, then SPnGX has a cn-network (resp. ck−network) of cardinality ≤ κ. [ABSTRACT FROM AUTHOR]

Let X be a metric continuum and n 2 N. Let Fn(X) be the hyperspace of nonempty subsets of X with at most n points. If 1 ≤ m < n, we consider the quotient space Fn m(X) = Fn(X)=Fm(X). Given a mapping f: X → X, we consider the induced mappings fn: Fn(X) → Fn(X) and fn m: Fn m (X) → Fn m (X). In this paper we study relations among the dynamics of the mappings f, fn and fn m and we answer some questions, by F. Barragán, A. Santiago-Santos and J. Tenorio, related to the properties: minimality, irreducibility, strong transitivity and turbulence. [ABSTRACT FROM AUTHOR]

In the present paper we introduce a generalization of the complete invariance property (CIP) for metric spaces, which we will call the ε-approximated complete invariance property (ε-ACIP). For our goals, we will use the so called degree of nondensifiability (DND) which, roughly speaking, measures (in the specified sense) the distance from a bounded metric space to its class of Peano continua. Our main result relates the ε-ACIP with the DND and, in particular, proves that a densifiable metric space has the ε-ACIP for each ε > 0. Also, some essentials differences between the CIP and the ε-ACIP are shown. [ABSTRACT FROM AUTHOR]

In this paper, we present a new class of cyclic (noncyclic) α and β-condensing operators and survey the existence of best proximity points (pairs) as well as coupled best proximity points (pairs) in the setting of reexive Busemann convex spaces. Then an application of the main existence result to study the existence of an optimal solution for a system of differential equations is demonstrated. [ABSTRACT FROM AUTHOR]