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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]

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]

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]

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]

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]

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]

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]

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]

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 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]

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 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]

This paper continues a series discussing aws in published assertions concerning fixed points in digital metric spaces. [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 a paper by Bella, Tokgös and Zdomskyy it is asked whether there exists a Tychonoff space X such that the remainder of Cp(X) in some compactification is Menger but not σ-compact. In this paper we prove that it is consistent that such space exists and in particular its existence follows from the existence of a Menger ultrafilter. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of Ɛ-contractive maps in fuzzy metric space (X,M,) and study the periodicities of Ɛ-contractive maps. We show that if (X,M,) is compact and f: X -! X is Ɛ-contractive, then P(f) = \1 n=1fn(X) and each connected component of X contains at most one periodic point of f, where P(f) is the set of periodic points of f. Furthermore, we present two examples to illustrate the applicability of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we describe the periodic points of automorphisms of a one dimensional solenoid, considering it as the inverse limit, lim k (S1, k) of a sequence ( k) of maps on the circle S1. The periodic points are discussed for a class of automorphisms on some higher dimensional solenoids also. [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]

Let I be a fuzzy metric interval and f be a continuous map from I to I. Denote by R(f), Δ(f) and ω(x, f) the set of recurrent points of f, the set of non-wandering points of f and the set of ω-limit points of x under f, respectively. Write ω(f) = U...ω(x, f), ωn+1(f) = ...ω(x, f) and Δn+1(f) = Δ(f:Δn(f)) for any positive integer n. In this paper, we show that Δ2(f) = R(f) and the depth of f is at most 2, and ω³(f) = ω²(f) and the depth of the attracting centre of f is at most 2. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study some topological distances properties, semidendrites and convexity on the symmetrized omega algebra. Furthermore, some properties and exponents on the symmetrized omega algebra are introduced. [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]

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]

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]

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]

In this paper, we prove a fixed point result for F-contraction principle in the framework of rectangular M-metric space. An example is also adopted to exhibit the utility of our result. Finally, we apply our fixed point result to show the existence of solution of Fredholm integral equation. [ABSTRACT FROM AUTHOR]

A space X is said to be set star-Lindelöf if for each nonempty subset A of X and each collection U of open sets in X such that Ā⊑⋃u, there is a countable subset V of U such that A ⊑ St(⋃V; u). The class of set star-Lindelöf spaces lie between the class of Lindelöf spaces and the class of star-Lindelöf spaces. In this paper, we investigate the relationship between set star-Lindelöf spaces and other related spaces by providing some suitable examples and study the topological properties of set star-Lindelöf spaces. [ABSTRACT FROM AUTHOR]

In this article, we prove that the group of all increasing homeomorphisms on R has exactly five normal subgroups, and the group of all homeomorphisms on R has exactly four normal subgroups. There are several results known about the group of homeomorphisms on R and about the group of increasing homeomorphisms on R ([2], [6], [7] and [8]), but beyond this there is virtually nothing in the literature concerning the topological structure in the aspects of topological dynamics. In this paper, we analyze this structure in some detail. [ABSTRACT FROM AUTHOR]

This paper introduces the ring of all real valued Baire one functions, denoted by B1(X) and also the ring of all real valued bounded Baire one functions, denoted by B*1 (X). Though the resemblance between C(X) and B1(X) is the focal theme of this paper, it is observed that unlike C(X) and C*(X) (real valued bounded continuous functions), B*1 (X) is a proper subclass of B1(X) in almost every non-trivial situation. Introducing B1-embedding and B*1-embedding, several analogous results, especially, an analogue of Urysohn's extension theorem is established. [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]

In this paper we study the dynamics of continuous endomorphisms on Polish groups. We offer necessary and sufficient conditions for a continuous endomorphism on a Polish group to be weakly mixing. We prove that any continuous endomorphism of an abelian Polish group can be extended in a natural way to a topologically mixing endomorphism on the countable infinite product of said group. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a new concept of KW-type m-contraction mapping. Then, we obtain some fixed point results for these mappings on M-metric spaces. Thus, we extend many well-known results for both single valued mappings and multivalued mappings such as the main results of Klim and Wardowski [13] and Altun et al. [4]. Also, we provide an interesting example to show the effectiveness of our result. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two new classes of nonlinear mappings and present some new existence and convergence theorems for these mappings in Banach spaces. More precisely, we employ the Krasnosel'ski iterative method to obtain fixed points of Suzuki-enriched nonexpansive mappings under different conditions. Moreover, we ap-proximate the fixed point of enriched-quasinonexpansive mappings via Ishikawa iterative method. [ABSTRACT FROM AUTHOR]

The current paper focuses on fundamental groups and Euler charac- teristics of various digital models of the 2-dimensional sphere. For all models that we consider, we show that the fundamental groups are trivial, and compute the Euler characteristics (which are not always equal). We consider the connected sum of digital surfaces and inves- tigate how this operation relates to the fundamental group and Euler characteristic. We also consider two related but different notions of a digital image having “no holes,” and relate this to the triviality of the fundamental group. Many of our results have origins in the paper [15] by S.-E. Han, which contains many errors. We correct these errors when possible, and leave some open questions. We also present some original results. [ABSTRACT FROM AUTHOR]

After the inception of the concept of a fuzzy metric by I. Kramosil and J. Michalek, and especially after its revision by A. George and G. Veeramani, the attention of many researches was attracted to the topology induced by a fuzzy metric. In most of the works devoted to this subject the resulting topology is an ordinary, that is a crisp one. Recently some researchers showed interest in the fuzzy-type topologies induced by fuzzy metrics. In particular, in the paper (J. J. Miñana, A. Šostak, Fuzzifying topology induced by a strong fuzzy metric, Fuzzy Sets and Systems 300 (2016), 24–39) a fuzzifying topology T : 2X → [0, 1] induced by a fuzzy metric m : X × X × [0,1) was constructed. In this paper we extend this construction to get the fuzzy topology T : [0, 1]X → [0, 1] and study some properties of this fuzzy topology. [ABSTRACT FROM AUTHOR]

In this paper we continue to investigate the impact that various separation axioms and covering properties have onto the cardinality of topological spaces. Many authors have been working in that field. To mention a few, let us refer to results by Arhangel'skii, Alas, Hajnal- Juhász, Bell-Gisburg-Woods, Dissanayake-Willard, Schroder and to the excellent survey by Hodel "Arhangel'ski?ı's Solution to Alexandroff's problem: A survey". Here we provide improvements and analogues of some of the results obtained by the above authors in the settings of more general separation axioms and cardinal invariants related to them. We also provide partial answer to Arhangel'skii's question concerning whether the continuum is an upper bound for the cardinality of a Hausdorff Lindelöf space having countable pseudo-character (i.e., points are Gδ). Shelah in 1978 was the first to give a consistent negative answer to Arhangel'skii's question; in 1993 Gorelic established an improved result; and further results were obtained by Tall in 1995. The question of whether or not there is a consistent bound on the cardinality of Hausdorff Lindelöf spaces with countable pseudo-character is still open. In this paper we introduce the Hausdorff point separating weight Hpw(X), and prove that (1) |X| ≤ Hpsw(X)aLc(X)ψ(X), for Hausdorff spaces and (2) |X| ≤ Hpsw(X)ωLc(X)ψ(X), where X is a Hausdorff space with a π-base consisting of compact sets with non-empty interior. In 1993 Schröder proved an analogue of Hajnal and Juhasz inequality |X| ≤ 2c(X)χ(X) for Hausdorff spaces, for Urysohn spaces by considering weaker invariant - Urysohn cellularity Uc(X) instead of cellularity c(X). We introduce the n-Urysohn cellularity n-Uc(X) (where n≥2) and prove that the previous inequality is true in the class of n-Urysohn spaces replacing Uc(X) by the weaker n-Uc(X). We also show that |X| ≤ 2Uc(X)πχ(X) if X is a power homogeneous Urysohn space. [ABSTRACT FROM AUTHOR]

In this paper, the full subcategory Hcomp of Top whose objects are Hausdorff compact spaces is identified as the orthogonal hull of the unit interval I = [0, 1]. The family of continuous maps rendered invertible by the reflector β ο ρ is deduced. [ABSTRACT FROM AUTHOR]

The dynamics of a projective transformation on a real projective space are studied in this paper. The two main aspects of these transforma-tions that are studied here are the topological entropy and the zeta function. Topological entropy is an inherent property of a dynami-cal system whereas the zeta function is a useful tool for the study of periodic points. We find the zeta function for a general projective transformation but entropy only for certain transformations on the real projective line. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new notion, named controlled shadowing property and we relate it to some notions in dynamical systems such as topological ergodicity, topologically mixing and specification properties. The relation between the controlled shadowing and chaos in sense of Li-Yorke is studied. At the end we give some examples to investigate the controlled shadowing property. [ABSTRACT FROM AUTHOR]

In this paper, we prove coincidence and common fixed points results under nonlinear contractions on a metric space equipped with an arbitrary binary relation. Our results extend, generalize, modify and unify several known results especially those are contained in Berzig [J. Fixed Point Theory Appl. 12, 221-238 (2012))] and Alam and Imdad [To appear in Filomat (arXiv:1603.09159 (2016))]. Interestingly, a corollary to one of our main results under symmetric closure of a binary relation remains a sharpened version of a theorem due to Berzig. Finally, we use examples to highlight the accomplished improvements in the results of this paper. [ABSTRACT FROM AUTHOR]