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Pretopology and multisets have an essential role in mathematics, especially, in some real-life applications. This paper aims to establish a recombination between multisets and pretopology. In addition, some new types of multi-pretopological structures are constructed and some of their properties are investigated. Multi-pre-open sets and multi-pre-closure operator are investigated. Moreover, for the isotonic multi-pretopological structures, and the multi-pre-minimal closed subsets are introduced and studied. Also, the concept of m-pre-connected structures are established as an application of given results and some types of strongly connected structures are investigated. Finally, the effect of multisets in medical information systems has been studied. [ABSTRACT FROM AUTHOR]

In a space X, if for every countable open cover U = { U n : n ∈ N } of X, we can find a subcover V = { U m k : k ∈ N } such that δ ({ m k : U m k ∈ V }) = 0 then the space is called a statistically compact space. Extending the recent works of Sarkar, Bal, and Rakshit on statistically compactness, we investigate statistically compactness of a topological space in the star-operator’s background. The concept of star statistically compactness is contrasted to other topological features. This study explains the attributes of star statistically compactness and its subspaces under diverse circumstances, especially under open continuous surjection. [ABSTRACT FROM AUTHOR]

AbstractLet R be a Cohen–Macaulay local ring. In this paper, we first describe the radicals of annihilators of stable categories of maximal Cohen–Macaulay R-modules. We then provide a necessary and sufficient condition for the Alexandrov topology of the stable category of maximal Cohen–Macaulay R-modules to be compact when R has an isolated singularity. Finally, we consider the case of a hypersurface of countable CM-representation type. [ABSTRACT FROM AUTHOR]

The soft fuzzy rough covering models were first proposed by Zhan and Sun (Comput Appl Math 38:149, 2019), which have garnered more interest in the last few years. Initially, they presented the notions of soft fuzzy neighborhood and the type k -approximation (for k = 1) based on the notion of soft fuzzy covering (i.e., soft fuzzy rough covering models using soft fuzzy neighborhoods). Then, Atef and Nada (Math Comput Simul 185:452–467, 2021) further offered type k -approximation (for k = 2 , 3 , 4) based on soft fuzzy covering (i.e., soft fuzzy rough covering models through soft fuzzy neighborhoods and soft fuzzy complement neighborhoods). However, as the notions of soft fuzzy neighborhood and soft fuzzy complement neighborhood are not able to ensure that they have the reflexive property, the type k -approximation (for each k ∈ { 1 , 2 , 3 , 4 } ) cannot satisfy the inclusion property (which is crucial for rough set models). The present paper will give an improvement to it in a very easy-to-understand manner so that the new type k -approximation has more desired properties ( k ∈ { 1 , 2 , 3 , 4 } ). We first define the notions of reflexive soft fuzzy neighborhood and reflexive soft fuzzy complement neighborhood, along with some key findings. Then, we present four new types of approximations based on soft fuzzy covering, each of which guarantees that a fuzzy subset to be approximated both includes its lower approximation and is included in its upper approximation (we call this inclusion property) apart from having usually ideal characteristics; several examples (including miner revisions) are also presented. Finally, a thorough explanation of a novel strategy that resolves the decision-making problem is provided through the new type k -approximations (k ∈ { 1 , 2 , 3 , 4 }) based on a family of soft fuzzy coverings, and a numerical example is given to show how applicable it is. [ABSTRACT FROM AUTHOR]

AbstractIn recent works, the concept of proximal groups was introduced as a generalization of topological groups. In this paper, we define the concept of quotient proximal groups with the help of canonical mappings, which is not really the generalization of quotient topological groups, in general. We study some topological properties of this quotient proximal group. We prove some important theorems namely induced mapping theorem, first isomorphism theorem and second isomorphism theorem for proximal groups. We also study numerous applications of first isomorphism theorem. The study is supported by various examples. [ABSTRACT FROM AUTHOR]

This article is devoted to present the degree to which a mapping defined from LX{L}^{X} to M,M, which is an (L,M)\left(L,M)-fuzzy bornology in the sense of Liang et al. In this respect, the degree to which LL-subset is bounded according to the given bornological structure is described. In addition, the notions of boundedness degree and the boundedness preserving degree of a mapping defined between bornological spaces are introduced. The characterizations of the provided notions are studied.

Binary relation plays a prominent role in the study of network structures. Since network structures are discrete system, topological techniques cannot be utilized to study the system. Rather, the concept of closed sets defined in closure spaces involving a closure operator is used in such types of studies. Recently, Šlapal and Pfaltz studied network structures by using foresets as defined in the Closure spaces via relation. In this paper, two types of closure operators via relations are taken into consideration to verify some results associated with networks. In one case, the relation is binary whereas in the other case the relation is reflexive and symmetric. The situation when a network structure is discrete/indiscrete is investigated. Further, a situation in which a set is closed in a tree-like network structure is obtained and examples in support are provided. [ABSTRACT FROM AUTHOR]

A (generalized) topological space is called an iso-dense space if the set of all its isolated points is dense in the space. The main aim of the article is to show in ZF a new characterization of iso-dense spaces in terms of special quasiorders. For a non-empty family A of subsets of a set X, a quasiorder ≲ A on X determined by A is defined. Necessary and sufficient conditions for A are given to have the property that the topology consisting of all ≲ A -increasing sets coincides with the generalized topology on X consisting of the empty set and all supersets of non-empty members of A . The results obtained, applied to the quasiorder ≲ D determined by the family D of all dense sets of a given (generalized) topological space, lead to a new characterization of non-trivial iso-dense spaces. Independence results concerning resolvable spaces are also obtained. [ABSTRACT FROM AUTHOR]

In this article, we develop innovative models of multigranulation rough fuzzy sets, utilizing fuzzy β -covering and incorporating a range of techniques such as fuzzy β -neighborhood, fuzzy complementary β -neighborhood, fuzzy β -minimal description, and fuzzy β -maximal description. The axiomatic characteristics of fuzzy β -neighborhoods within the context of fuzzy β -covering based multigranulation rough fuzzy sets (F β CMGRFS) are analyzed. Thus, we introduce seven new classes of F β CMGRFS and investigate their relevant properties. Furthermore, the connections and associations between these techniques are established. Thus, we provide a set of observations and propositions that highlight the value and dissimilarities of our proposed models compared to others. A test example is produced to verify the applicability of the presented strategies and treat MCGDM issues. Finally, we compare the outcomes of our approaches and the existing studies to check the reliability and the validity of our work. [ABSTRACT FROM AUTHOR]

In this paper, new purely topological approaches are furnished in order to characterize self-majorizing elements in an Archimedean vector lattice A. More precisely, it is shown that an element 0 < f ∈ A is a self-majorizing element if and only if every f-maximal order ideal of A is relatively uniformly closed. In addition, it is proved that self-majorizing elements are characterized via the hull–kernel topology on both the set of all proper prime order ideals P and on the set of all g-maximal order ideals Q of A, for all g ∈ A +. In fact, the set of all prime order ideals of A not containing f (respectively, the set of all g-maximal order ideals of A not containing f, for all g ∈ A +) is a closed with respect to the hull–kernel topology on P (respectively, on Q) if and only if f is a self-majorizing element in A. [ABSTRACT FROM AUTHOR]

We employ closure operators associated with n-ary relations, n > 1 an integer, to provide the digital space ℤ3 with connectedness structures. We show that each of the six inscribed tetrahedra obtained by canonical tessellation of a digital cube in ℤ3 with edges consisting of 2n − 1 points is connected. This result is used to prove that certain bounding surfaces of the polyhedra in ℤ3 that may be face-to-face tiled with such tetrahedra are digital Jordan surfaces (i.e., separate ℤ3 into exactly two connected components). An advantage of these Jordan surfaces over those with respect to the Khalimsky topology is that they may possess acute dihedral angles π 4 while, in the case of the Khalimsky topology, the dihedral angles may never be less than π 2 . [ABSTRACT FROM AUTHOR]

In this paper, we focus our attention on an outer Lebesgue measure and density-type generalized topologies connected with this measure and with nondecreasing and unbounded sequences of positive reals. Some properties of such generalized topologies and continuous functions connected with this space are presented. [ABSTRACT FROM AUTHOR]

We introduce the concept of generalized suprametric spaces, which subsumes some existing abstract metric spaces. Then, we show the existence of fixed points for maps satisfying nonlinear contractions involving either extended comparison or ρ\rho -subhomogeneous functions. This study was carried out in generalized suprametric spaces as well as partially ordered generalized suprametric spaces. Some related results in JS-metric spaces and in bb-suprametric spaces are improved or extended.

This study investigates some topological properties of locally semi-compact Ir-topological groups and establishes the relationship between Ir-topological groups and semi-compact spaces. The proved theorems generalize the corresponding results of Ir-topological group. Finally, we define a quotient topology on the Ir-topological group and study some topological properties of the space.

The paper initially investigates semi-topological properties of the Khalimsky (K-, for brevity) topological version of the classical Jordan curve theorem. Assume a simple closed K-curve with l elements in (Z 2 , κ 2) that is the K-topological plane, denoted by C K l for brevity. Then every C K l separates ( Z 2 , κ 2) into exactly two nonempty components that may be neither open nor closed in ( Z 2 , κ 2) . Hence we need to investigate semi-topological features of C K l and Z 2 \ C K l in (Z 2 , κ 2) . We first show that not every C K l is always semi-open or semi-closed in (Z 2 , κ 2) . Second, we find a condition for C K l to be either semi-open or semi-closed in (Z 2 , κ 2) . After establishing a continuous analog of C K l denoted by A (C K l) (⊂ R 2) , we show that A (C K l) is both semi-open and semi-closed in (R 2 , U) that is the 2-dimensional real plane with the usual topology. Besides, we show that A (C K l) always separates (R 2 , U) into two non-empty components, denoted by C and D, that are both semi-open and semi-closed to obtain a partition of R 2 , i.e., { C , D , A (C K l) } . Finally, we obtain a partition of Z 2 , i.e., { I (C K l) , O (C K l) , C K l } and prove that each of I (C K l) and O (C K l) is semi-closed and it need not be semi-open, where I (C K l) and O (C K l) are called an inside and outside of C K l , respectively. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of large scale resemblance structure as a new large scale structure by axiomatizing the concept of being alike in large scale for a family of subsets of a set. We see that in a particular case, large scale resemblances on a set can induce a nearness on it, and as a consequence, we offer a relatively big class of examples to show that not every near family is contained in a bunch. Besides, We show how some large scale properties like asymptotic dimension can be generalized to large scale resemblance spaces. [ABSTRACT FROM AUTHOR]

The properties of generalized topological spaces generated by some kind of density operators are presented. Moreover, the separation axioms of such generalized topological spaces are investigated. The examples of generalized topological spaces satisfing all separation axioms are included. [ABSTRACT FROM AUTHOR]

In this article, we introduce s-s-γ\gamma cover using the notion of star operator, which is an extension of the previous results on s-γ\gamma covers. We also define s-s-dense set to address the issue that a statistical dense subset of an s-s-γ\gamma cover is not an s-s-γ\gamma cover. Furthermore, we study some results on s-γ\gamma covers in subspace topology.

The cluster soft point is an attempt to introduce a novel generalization of the soft closure point and the soft limit point. A cluster soft set is defined to be the system of all cluster soft points of a soft set. Then, the fundamental properties of cluster soft sets are demonstrated. Moreover, the concept of a cluster soft topology on a universal set is introduced with regard to cluster soft sets. The cluster soft topology is derived from a soft topology with an associated soft ideal, but it is finer than the original soft topology. On the other hand, if we start constructing the cluster soft topology from another cluster soft topology, we will end up with the first cluster soft topology we started with. The implication of cluster soft topologies is highlighted using some examples. Eventually, we represent the cluster soft closed sets in terms of several forms of soft sets. [ABSTRACT FROM AUTHOR]

In this paper we work in the category of approach spaces with contractions [14], the objects of which are sets endowed with a numerical distance between sets and points. Approach spaces are to be considered a simultaneous generalization of both quasi-metric and topological spaces. Especially the fundamental notion of distance is reminiscent of the closure operator in a topological space and of the point-to-set distance in a quasi-metric space. The embedding of the category of topological spaces with continuous maps and of quasi-metric spaces with non-expansive maps is extremely nice. Every approach space has both a quasi-metric coreflection as well as a topological coreflection. Different approach spaces though can have the same topological as well as the same quasi-metric coreflection, in other words, in general these coreflections do not determine the approach space. In this paper we investigate approach spaces for which these coreflections, do determine the approach space. We will call such spaces supported. We prove that in the setting of compact approach spaces many examples of supported approach spaces can be found. Thus, compact spaces that are base-regular, which is a weakening of regularity, are always supported. An important feature of supported approach spaces is the behaviour of contractions. On a supported domain contractivity is characterized by the combination of continuity for the topological coreflection and non-expansiveness for the quasi-metric coreflection. This result implies that a supported approach space actually is the infimum of its quasi-metric and its topological coreflection. In the course of our study we also give several more examples of both supported and non-supported approach spaces. [ABSTRACT FROM AUTHOR]

Given a monoid S, each S-set may be endowed in a natural way with an Alexandroff topology. Furthermore, if a S-congruence is given on some S-set, we can also endow the corresponding quotient set with an Alexandroff topology. Bearing this in mind, for any monoid S admitting a countably infinite descending chain, we are able to define a specific quotient S-set which is the object of our investigation. More in detail, we carry out a detailed study using several properties of the S-orbits, and we first prove that these quotients are covered by a subcollection of closed subsets related to a suitable notion of dependence on union of subsets. Secondly, we characterize the noetherianity of these quotients in terms of the noetherianity of the monoid S and, finally, we focus our attention to two specific kinds of descending and ascending chains, analyzing some of their main properties on general Alexandroff spaces and, next, showing that the ascending chain stabilizes strictly before than the descending one in the case of our quotients. [ABSTRACT FROM AUTHOR]

In this manuscript, we first generate topological structures by subset neighborhoods and ideals and apply to establish some generalized rough-set models. Then, we present other types of generalized rough-set models directly defined by the concepts of subset neighborhoods and ideals. We explore the main characterizations of the proposed approximation spaces and compare them in terms of approximation operators and accuracy measures. The obtained results and given examples show that the second type of the proposed approximation spaces is better than the first one in cases of uu and ⟨u⟩\langle u\rangle , whereas the relationships between the rest of the six cases are posted as an open question. Moreover, we demonstrate the advantages of the current models to decrease the upper approximation and increase the lower approximation compared to the existing approaches in published literature. Algorithms and a flow chart are given to illustrate how the exact and rough sets are determined for each approach. Finally, we analyze the information system of dengue fever to confirm the efficiency of our approaches to maximize the value of accuracy and shrink the boundary regions.

The main objective of this study is to define a new class of bipolar soft (BS) separation axioms known as BS T˜˜i{\widetilde{\widetilde{T}}}_{i}-space (i=0,1,2,3,4)\left(i=0,1,2,3,4). This type is defined in terms of ordinary points. We prove that BS T˜˜i{\widetilde{\widetilde{T}}}_{i}-space implies BS T˜˜i−1{\widetilde{\widetilde{T}}}_{i-1}-space for i=1,2i=1,2; however, the opposite is incorrect, as demonstrated by an example. For i=0,1,2,3,4i=0,1,2,3,4, we investigate that every BS T˜˜i{\widetilde{\widetilde{T}}}_{i}-space is soft T˜i{\widetilde{T}}_{i}-space; and we set up a condition in which the reverse is true. Moreover, we point out that a BS subspace of a BS T˜˜i{\widetilde{\widetilde{T}}}_{i}-space is a BS T˜˜i{\widetilde{\widetilde{T}}}_{i}-space for i=0,1,2,3i=0,1,2,3.

Let I C L (n) be the number of clopen topologies on a finite set of n elements. It is proved that the explicit formula for finding the total number of clopen topologies is ∑ k = 1 n S n , k = ∑ i = 1 n CL (n , 2 i) , where S n , k is the Stirling number of the second kind and CL (n , 2 i) is the number of clopen topologies having 2 i open sets, i = 1 , 2 , 3 ,... , n . Some results concerning the number of clopen topological spaces whose topologies have the same cardinality are also obtained. [ABSTRACT FROM AUTHOR]

The intention of this article is to introduce and characterise the concept of preopen sets and prelocally closed sets in Generalised Topology and Minimal structure spaces.

In this paper, we develop an approach to the presentation of set-theoretic topology based on the systematic use of two standard operations: the union operation and the closure operation of a collection of sets (i.e., the closure with respect to the union operation). Definitions of these operations are given and their basic properties are formulated and proved. Based on the results obtained, we present the foundations of elementary topology. [ABSTRACT FROM AUTHOR]

We study closure operators on complete lattices and define for them the concepts of closed and open bases as well as that of a neighborhood base. Some properties of these concepts are studied including relationships between them. In particular, we find sufficient conditions under which the properties are analogous to those known for topological spaces. [ABSTRACT FROM AUTHOR]

Sarsak [M. S. Sarsak, On some properties of generalized open sets in generalized topological spaces, Demonstr. Math. 46 (2013), no. 2, 415–427] studied some properties of generalized open sets in generalized topological spaces (GTSs); the primary purpose of this article is to investigate more properties of generalized open sets in GTSs. We mainly study the behaviours of regular closed sets, semi-open sets, regular semi-open sets, preopen sets, and β\beta -open sets in GTSs analogous to their behaviours in topological spaces.

We discuss certain closure operators that generalize the Alexandroff topologies. Such a closure operator is defined for every ordinal α>0\alpha \gt 0 in such a way that the closure of a set AA is given by closures of certain α\alpha -indexed sequences formed by points of AA. It is shown that connectivity with respect to such a closure operator can be viewed as a special type of path connectivity. This makes it possible to apply the operators in solving problems based on employing a convenient connectivity such as problems of digital image processing. One such application is presented providing a digital analogue of the Jordan curve theorem.

Quite recently, a new minimal structure mH⋆{m}_{H}^{\star } and an mnmn-ℋg{{\mathcal{ {\mathcal H} }}}_{g}-closed set have been introduced in a previous study [T. Noiri and V. Popa, Closed sets in hereditary bi m-spaces, Questions Answers General Topol. 38 (2020), 133–142] by using two minimal structures m, nn and a hereditary class ℋ{\mathcal{ {\mathcal H} }}. In this paper, we introduce and investigate the notions of (m,n)\left(m,n)-ℋg⋆{{\mathcal{ {\mathcal H} }}}_{g}^{\star }-regularity and (m,n)\left(m,n)-ℋg⋆{{\mathcal{ {\mathcal H} }}}_{g}^{\star }-normality in a hereditary bi m-space (X,m,n,ℋ)\left(X,m,n,{\mathcal{ {\mathcal H} }}).

In the paper, we apply some of the results from the theory of ball spaces in semimetric setting. This allows us to obtain fixed point theorems which we believe to be unknown to this day. As a byproduct, we obtain the equivalence of some different notions of completeness in semimetric spaces where the distance function is 1-continuous. In the second part of the article, we generalize the Caristi–Kirk results for b-metric spaces. Additionally, we obtain a characterization of semicompleteness for 1-continuous b-metric spaces via a fixed point theorem analogous to a result of Suzuki. In the epilogue, we introduce the concept of convergence in ball spaces, based on the idea that balls should resemble closed sets in topological sets. We prove several of its properties, compare it with convergence in semimetric spaces and pose several open questions connected with this notion. [ABSTRACT FROM AUTHOR]

A new topology is associated with the collection of edges in each digraph, we introduce a new topology for digraphs. The independent compatible edges topology is the name given to this topology. The independent compatible edges topology is constructed using a sub basis collection on the set of edges E. Then, we examine various properties and explain the independent compatible edge topology of a few key graph types. A few of the topology's initial outcomes were examined. Our goal is to begin understanding some characteristics of directed graphs by examining the ability compatible edges topology, which is addressed in this article. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to introduce and study the notions of -set and -continuity via biopeation. We discuss some properties of these notions. [ABSTRACT FROM AUTHOR]

In 1994, S.G. Matthews introduced the notions of partial metric and weighted quasi-metric and, in addition, he showed that both are dual. In fact, he developed a method for generating weightable quasi-metrics from partial metrics and conversely. Recently, J.J. Miñana and O. Valero introduced a general method with the aim of inducing quasi-metrics from partial metrics via real-valued functions in such a way that the original Matthews method can be retrieved as a special case when an appropriate function is chosen. However, they showed that the aforementioned method does not induce in general weighted quasi-metrics and, thus, they posed the question about what properties must be fulfilled by this type of functions in order to guarantee that the generated quasi-metric is weighted. In this paper, we answer such a question providing a characterization of such functions through the use of what we have called weight difference function and a new type of quadrangular triplets. Finally, a new characterization of weightable quasi-metric spaces is also given and a few properties of weightable quasi-metric spaces are discussed. [ABSTRACT FROM AUTHOR]

We introduce the concept of suprametric space and study some basic properties of its topology. Then we show that certain contraction maps in suprametric spaces have a unique fixed point either the space is complete or it contains a nonempty ω -limit set. Next, we construct three suprametrics in partially ordered vector spaces, and utilize them to derive several fixed point theorems. Finally, we apply the obtained results to investigate the existence of solutions to some nonlinear integral and matrix equations. [ABSTRACT FROM AUTHOR]

Sarsak [On μ\mu -compact sets in μ\mu -spaces, Questions Answers Gen. Topology 31 (2013), no. 1, 49–57] introduced and studied the class of μ\mu -Lindelöf sets in μ\mu -spaces. Mustafa [μ\mu -semi compactness and μ\mu -semi Lindelöfness in generalized topological spaces, Int. J. Pure Appl. Math. 78 (2012), no. 4, 535–541] introduced and studied the class of μ\mu -semi-Lindelöf sets in generalized topological spaces (GTSs); the primary purpose of this paper is to investigate more properties and mapping properties of μ\mu -semi-Lindelöf sets in μ\mu -spaces. The class of μ\mu -semi-Lindelöf sets in μ\mu -spaces is a proper subclass of the class of μ\mu -Lindelöf sets in μ\mu -spaces. It is shown that the property of being μ\mu -semi-Lindelöf is not monotonic, that is, if (X,μ)\left(X,\mu ) is a μ\mu -space and A⊂B⊂XA\subset B\subset X, where AA is μB{\mu }_{B}-semi-Lindelöf, then AA need not be μ\mu -semi-Lindelöf. We also introduce and study a new type of generalized open sets in GTSs, called ωμ{\omega }_{\mu }-semi-open sets, and investigate them to obtain new properties and characterizations of μ\mu -semi-Lindelöf sets in μ\mu -spaces.

In this article, we exploit the relations of total belong and total non-belong to introduce new soft separation axioms with respect to ordinary points, namely tttt-soft pre Ti(i=0,1,2,3,4){T}_{i}\hspace{0.33em}\left(i=0,1,2,3,4) and tttt-soft pre-regular spaces. The motivations to use these relations are, first, cancel the constant shape of soft pre-open and pre-closed subsets of soft pre-regular spaces, and second, generalization of existing comparable properties on classical topology. With the help of examples, we show the relationships between them as well as with soft pre Ti(i=0,1,2,3,4){T}_{i}\hspace{0.33em}\left(i=0,1,2,3,4) and soft pre-regular spaces. Also, we explain the role of soft hyperconnected and extended soft topological spaces in obtaining some interesting results. We characterize a tttt-soft pre-regular space and demonstrate that it guarantees the equivalence of tttt-soft pre Ti(i=0,1,2){T}_{i}\hspace{0.33em}\left(i=0,1,2). Furthermore, we investigate the behaviors of these soft separation axioms with the concepts of product and sum of soft spaces. Finally, we introduce a concept of pre-fixed soft point and study its main properties.

The key concept in this research is blended vertices systems, which are used to introduce and investigate the B-space. The equivalence of two reflexive B-spaces is superimposed on the significant and sufficient station. In addition, the concepts of b-derived graphs, b-open graphs, b-closed graphs, b-interior operators, b-closure operator, and B-subspace are discussed. A unique supra topological space is introduced from a B-space. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study and investigate the concept of weak topological BCK-algebras which is obtained by combining a certain topology to a BCK-algebra that makes the binary operation defined on the BCK-algebra compatible. Several properties of weak topological BCK-algebras are investigated. Continuity between weak topological BCK-algebra is studied and a new topology is constructed by using special subsets of a BCK-algebra and further results are obtained. [ABSTRACT FROM AUTHOR]

Some sufficient conditions are presented for the continuity of the mapping f : ⟨ X , τ X * ⟩ → ⟨ Y , τ Y * ⟩ , where τ X * and τ Y * are the topologies induced by the local function on 푋 and 푌, resp. under the assumption that the mapping from ⟨ X , τ X ⟩ to ⟨ Y , τ Y ⟩ is continuous. Further, we consider open and closed functions in the cases in which the open (or closed) mapping is preserved through the domain and codomain "idealization". Through several examples, we illustrate that the considered conditions cannot be weakened. [ABSTRACT FROM AUTHOR]