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In this paper, we introduce the strong b-suprametric spaces in which we prove the fixed point principles of Banach and Edelstein. Moreover, we prove a variational principle of Ekeland and deduce a Caristi fixed point theorem. Furthermore, we introduce the strong b-supranormed linear spaces in which we establish the fixed point principles of Brouwer and Schauder. As applications, we study the existence of solutions to an integral equation and to a third-order boundary value problem. [ABSTRACT FROM AUTHOR]

The aim of this paper is to construct ultrafilters without immediate predecessors in the Rudin-Frolik order in β κ \ κ , where κ is a regular cardinal. This generalizes the problem posed by Peter Simon more than 40 years ago. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to generalize the measure of non-compactness for the space of continuous functions over the T3½ space. Motivated by the generalized Arzelà-Ascoli theorem for Tichonoff space T via Wallman compactifiaction Wall(T), we constuct a measure of non-compactness for the space Cb(T). We also study some of the properties of this object and give another version of Darbotype theorem, suitable for this particular case. [ABSTRACT FROM AUTHOR]

AbstractA J-frame is a pointfree topology counterpart of the concept of a J-space. In this paper, we present a study of some frames satisfying conditions that are stronger (and some that are weaker) than those defining J-frames. Conditions under which these frames coincide with J-frames are established - pointing out similarities and differences among these frames in the process. Moreover, JC-frames are introduced by mimicking the defining conditions of a J-frame, modulo replacing “compact” with “connected.” We show that all connected frames are JC-frames, not conversely. Any frame that can be written as a join of two disjoint connected sublocales is a JC-frame. Upon recalling that a frame with no points is a J-frame if and only if it is connected, we infer that all non-spatial J-frames are JC-frames. [ABSTRACT FROM AUTHOR]

The system S c (L) consisting of joins of closed sublocales of a locale L is known to be a frame, and for L subfit it coincides with the Booleanization S b (L) of the coframe of sublocales of L. In this paper, we study S b (L) for a general locale L. We show that S c (L) is always a subframe of S b (L) . Moreover, if X is a T D -space, we prove that S b (Ω (X)) is precisely the set of classical subspaces of X, and that a locale L is T D -spatial iff the Boolean algebra S b (L) is atomic. Some functoriality properties of S b (L) are also studied. [ABSTRACT FROM AUTHOR]

This paper constructs the compactification of rough pseudo metric spaces using the concept of Čech rough proximity spaces, in terms of rough linked grills which are generalizations of Cauchy sequences. [ABSTRACT FROM AUTHOR]

In this paper, we study proper left invariant metrics of hypergroups and show that any two proper left invariant metrics on a hypergroup, are coarsely equivalent. [ABSTRACT FROM AUTHOR]

There exist two known types of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them (Goranko in Filter and ultrafilter extensions of structures: universal-algebraic aspects, preprint, 2007) comes from modal logic and universal algebra, and in fact goes back to Jónsson and Tarski (Am J Math 73(4):891–939, 1951; 74(1):127–162, 1952). Another one (Saveliev in Lect Notes Comput Sci 6521:162–177, 2011; Saveliev in: Friedman, Koerwien, Müller (eds) The infinity project proceeding, Barcelona, 2012) comes from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups (Hindman and Strauss in Algebra in the Stone–Čech Compactification, W. de Gruyter, Berlin, 2012) as its main precursor. By a classical fact of general topology, the space of ultrafilters over a discrete space is its largest compactification. The main result of Saveliev (Lect Notes Comput Sci 6521:162–177, 2011; in: Friedman, Koerwien, Müller (eds) The infinity project proceeding, Barcelona, 2012), which confirms a canonicity of this extension, generalizes this fact to discrete spaces endowed with an arbitrary first-order structure. An analogous result for the former type of ultrafilter extensions was obtained in Saveliev (in On two types of ultrafilter extensions of binary relations. arXiv:2001.02456). Results of such kind are referred to as extension theorems. After a brief introduction, we offer a uniform approach to both types of extensions based on the idea to extend the extension procedure itself. We propose a generalization of the standard concept of first-order interpretations in which functional and relational symbols are interpreted rather by ultrafilters over sets of functions and relations than by functions and relations themselves, and define ultrafilter models with an appropriate semantics for them. We provide two specific operations which turn ultrafilter models into ordinary models, establish necessary and sufficient conditions under which the latter are the two canonical ultrafilter extensions of some ordinary models, and obtain a topological characterization of ultrafilter models. We generalize a restricted version of the extension theorem to ultrafilter models. To formulate the full version, we propose a wider concept of ultrafilter models with their semantics based on limits of ultrafilters, and show that the former concept can be identified, in a certain way, with a particular case of the latter; moreover, the new concept absorbs the ordinary concept of models. We provide two more specific operations which turn ultrafilter models in the narrow sense into ones in the wide sense, and establish necessary and sufficient conditions under which ultrafilter models in the wide sense are the images of ones in the narrow sense under these operations, and also are two canonical ultrafilter extensions of some ordinary models. Finally, we establish three full versions of the extension theorem for ultrafilter models in the wide sense. The results of the first three sections of this paper were partially announced in Poliakov and Saveliev (in: Kennedy, de Queiroz (eds) On two concepts of ultrafilter extensions of first-order models and their generalizations, Springer, Berlin, 2017). [ABSTRACT FROM AUTHOR]

The purpose of the paper is to continue the study of completeness for symmetric spaces. We study variants of completeness in symmetric spaces and present some properties regarding completeness in symmetric spaces. Finally, as an application of completeness we will present the fixed point result for multivalued maps. [ABSTRACT FROM AUTHOR]

In this paper, we introduce coproducts of proximity spaces. After exploring several of their basic properties, we show that given a collection of proximity spaces, the coproduct of their Smirnov compactifications proximally and densely embeds in the Smirnov compactification of the coproduct of the original proximity spaces. We also show that the dense proximity embedding is a proximity isomorphism if and only if the index set is finite. After constructing a number of examples of coproducts and their Smirnov compactifications, we explore several properties of the Smirnov compactification of the coproduct, including its metrizability, connectedness of the boundary, dimension, and its relation to the Stone–Cech compactification. In particular, we show that the Smirnov compactification of the infinite coproduct is never metrizable and that its boundary is highly disconnected. We also show that the proximity dimension of the Smirnov compactification of the coproduct equals the supremum of the covering dimensions of the individual Smirnov compactifications and that the Smirnov compactification of the coproduct is homeomorphic to the Stone–Cech compactification if and only if each individual proximity space is equipped with the Stone–Cech proximity. We finish with an example of a coproduct with the covering dimension 0 but the proximity dimension ∞ . [ABSTRACT FROM AUTHOR]

Given a ring R and one of its proper ideals S, we obtain a compactification of the prime spectrum of S through a mainly algebraic process. We call it the R-nilcompactification of Spec S . We study some categorical properties of this construction. [ABSTRACT FROM AUTHOR]

Quasi-nearness biframes provide an asymmetric setting for the study of nearness; in Frith and Schauerte (Quaest Math 33:507–530, 2010 ) a completion (called a quasi-completion) was constructed for such structures and in Frith and Schauerte (Quaest Math, 2012 ) completeness was characterized in terms of the convergence of regular Cauchy bifilters. In this paper questions of functoriality for this quasi-completion are considered and one sees that having enough regular Cauchy bifilters plays an important rôle. The quasi-complete strong quasi-nearness biframes with enough regular Cauchy bifilters are seen to form a coreflective subcategory of the strong quasi-nearness biframes with enough regular Cauchy bifilters. Here a significant difference between the symmetric and asymmetric cases emerges: a strong (even quasi-uniform) quasi-nearness biframe need not have enough regular Cauchy bifilters. The Cauchy filter quotient leads to further characterizations of those quasi-nearness biframes having enough regular Cauchy bifilters. The fact that the Cauchy filter quotient of a totally bounded quasi-nearness biframe is compact shows that any totally bounded quasi-nearness biframe with enough regular Cauchy bifilters is in fact quasi-uniform. The paper concludes with various examples and counterexamples illustrating the similarities and differences between the symmetric and asymmetric cases. [ABSTRACT FROM AUTHOR]

For a metrizable topological space X it is well known that in general the Čech-Stone compactification β(X) or the Wallman compactification W(X) are not metrizable. To remedy this fact one can alternatively associate a point-set distance to the metric, a so called approach distance. It is known that in this setting both a Čech-Stone compactification β*(X) and a Wallman compactification W*(X) can be constructed in such a way that their approach distances induce the original approach distance of the metric on X [23], [24]. The main goal in this paper is to formulate necessary and sufficient conditions for an approach space X such that the Čech-Stone compactification β*(X) and the Wallman compactification W*(X) are isomorphic, thus answering a question first raised in [24]. The first clue to reach this goal is to settle a question left open in [11], to formulate sufficient conditions for a compact approach space to be normal. In particular the result shows that the Čech-Stone compactification β*(X) of a uniform T2 space, is always normal. We prove that the Wallman compactification W*(X) is normal if and only if X is normal, and we produce an example showing that, unlike for topological spaces, in the approach setting normality of X is not sufficient for β*(X) and W*(X) to be isomorphic. We introduce a strengthening of the regularity condition on X, which we call ideal-regularity, and in our main theorem we conclude that X is ideal-regular, normal and T1 if and only if X is a uniform T1 approach space with β*(X) and W*(X) isomorphic. Classical topological results are recovered and implications for (quasi-)metric spaces are investigated. [ABSTRACT FROM AUTHOR]

In this paper we consider two questions in the realm of paratopological groups: When does multiplication on a given Tychonoff paratopological group H admit an extension to continuous multiplication on the Dieudonné completion, $$\mu {H}$$ , of H in such a way that H turns into a dense subgroup of the paratopological group $$\mu {H}$$ ? and, if A and B are bounded subsets of paratopological groups G and H, respectively, is $$A\times B$$ bounded in $$G\times H$$ ? The motivation for these questions comes from the field of topological groups and they are receiving a special attention as an interesting source of open problems. [ABSTRACT FROM AUTHOR]

For a topological zero-dimensional Hausdorff space (X,) it is well known that the Banaschewski compactification ζ(X,) is of Wallman-Shanin-type, meaning that there exists a closed basis (the collection of all clopen sets), such that the Wallman-Shanin compactification with respect to this closed basis is isomorphic to ζ(X,). For an approach space (X,) the Wallman-Shanin compactification W (X,) with respect to a Wallman-Shanin basis (a particular basis of the lower regular function frame) was introduced by R. Lowen and the second author. Recently, various constructions of the Banaschewski compactification known for a topological space were generalised to the approach case. Given a Hausdorff zero-dimensional approach space (X,), constructions of the Banaschewski compactification ζ∗(X,) were developed by the authors. In this paper we construct a particular Wallman-Shanin basis for (X,) and show that the Wallman-Shanin compactification with respect to this particular basis is isomorphic to ζ∗(X,). [ABSTRACT FROM AUTHOR]

Generally, the term uc-ness means some continuity is uniform. A metric space X is uc when any continuous function from X to [0, 1] is uniformly continuous and a metrizable space X is a Nagata space when it can be equipped with a uc metric. We consider natural forms of uc-ness for the $${\omega_\mu}$$ -metric spaces, which fill a very large and interesting class of uniform spaces containing the usual metric ones, and extend to them various different formulations of the metric uc-ness, by additionaly proving their equivalence. Furthermore, since any $${\omega_\mu}$$ -compact space is uc and any uc $${\omega_\mu}$$ -metric space is complete, in the line of constructing dense extensions which preserve some structure, such as uniform completions, we focus on the existence for an $${\omega_\mu}$$ -metrizable space of dense topological extensions carrying a uc $${\omega_\mu}$$ -metric. In this paper we show that an $${\omega_\mu}$$ -metrizable space X is uc-extendable if and only if there exists a compatible $${\omega_\mu}$$ -metric d on X such that the set X′ of all accumulation points in X is crowded, i.e., any $${\omega_\mu}$$ -sequence in X′ has a d-Cauchy $${\omega_\mu}$$ -subsequence in X′. [ABSTRACT FROM AUTHOR]

For each poset P, we construct a polytope A (P) called the P-associahedron. Similarly to the case of graph associahedra, the faces of A (P) correspond to certain nested collections of subsets of P. The Stasheff associahedron is a compactification of the configuration space of n points on a line, and we recover A (P) as an analogous compactification of the space of order-preserving maps P → R . Motivated by the study of totally nonnegative critical varieties in the Grassmannian, we introduce affine poset cyclohedra and realize these polytopes as compactifications of configuration spaces of n points on a circle. For particular choices of (affine) posets, we obtain associahedra, cyclohedra, permutohedra, and type B permutohedra as special cases. [ABSTRACT FROM AUTHOR]

We extend the notion of local connectedness and uniform local connectedness to the category MσFrm of metric σ-frames. We provide the construction of the uniformly locally connected reflection of a locally connected metric σ-frame. We also discuss complete metric σ-frames and show the existence of a completion for a metric σ-frame in the category MσFrm. [ABSTRACT FROM AUTHOR]

In some previous papers we studiedgeneralizednearness by supernear spaces in connection withunificationsand topologicalextensions, respectively. Now, this last part of series deals with those supernear spaces, whichcoverclosely LODATO spaces, b-supertopologies and hence EF-proximities as well. Moreover, we will show that some special kind of these spaces have alocaltopological extension iff they aresuperbunchspaces. Consequently, thisfundamentalresult leading us to a furthergeneralizationof LODATO's theorem, and in addition Doîtchinov's achievement can be dealt with, but in a modified fashion. [ABSTRACT FROM AUTHOR]

The frame Sc(L) generated by closed sublocales of a locale L is known to be a natural Boolean ("discrete") extension of a subfit L; also it is known to be its maximal essential extension. In this paper we first show that it is an essential extension of any L and that the maximal essential extensions of L and Sc(L) are isomorphic. The construction Sc is not functorial; this leads to the question of individual liftings of homomorphisms L → M to homomorphisms Sc(L) → Sc(M). This is trivial for Boolean L and easy for a wide class of spatial L, M. Then, we show that one can lift all h : L → 2 for weakly Hausdorff L (and hence the spectra of L and Sc(L) are naturally isomorphic), and finally present liftings of h : L → M for regular L and arbitrary Boolean M. [ABSTRACT FROM AUTHOR]

We introduce a notion of connectedness as a generalization of connectedness in its usual sense. We find counterparts for most existing basic results on connectedness in this new setting. [ABSTRACT FROM AUTHOR]

Amenable groups have a close relationship with various mathematical concepts. In this paper we consider the correlation between the amenability and semigroup compactification of a locally compact group. It is observed that in the case that G is an amenable group, there are some wonderful properties for G LUC . Motivated by the definition provided by N. Hindman and D. Strauss, we defined the set L I M 0 ( G ) and other sets such as Φ ( G ) and Δ ⁎ ( G ) to achieve the desired results and we considered the correlation between the amenability and semigroup compactification of a locally compact group. [ABSTRACT FROM AUTHOR]

A space is reversible if every continuous bijection of the space onto itself is a homeomorphism. In this paper we study the question of which countable spaces with a unique non-isolated point are reversible. By Stone duality, these spaces correspond to closed subsets in the Čech–Stone compactification of the natural numbers βω . From this, the following natural problem arises: given a space X that is embeddable in βω , is it possible to embed X in such a way that the associated filter of neighborhoods defines a reversible (or non-reversible) space? We give the solution to this problem in some cases. It is especially interesting whether the image of the required embedding is a weak P -set. [ABSTRACT FROM AUTHOR]

It is known that every remainder of a topological group is Lindelöf or pseudocompact. Motivated by this result, we study in this paper the question when a topological group G has a normal remainder. Under mild conditions on G we show that under the Continuum Hypothesis, if the Čech–Stone remainder G ⁎ of G is normal, then it is Lindelöf. [ABSTRACT FROM AUTHOR]

In this paper, we give a useful criterion for the existence of reflections. Based on this criterion, we obtain a number of results about topological free objects, completions, compact reflections and ( ω -)totally bounded reflections. [ABSTRACT FROM AUTHOR]

A finite sequence a → = 〈 a i 〉 i = 1 m in Z ∖ { 0 } is compressed provided a i ≠ a i + 1 for i < m . It is known that if a → = 〈 a i 〉 i = 1 m and b → = 〈 b i 〉 i = 1 k are compressed sequences in Z ∖ { 0 } , then there exist idempotents p and q in β Q d ∖ { 0 } such that a 1 p + a 2 p + … + a m p = b 1 q + b 2 q + … + b k q if and only if b → is a rational multiple of a → . In fact, if b → is not a rational multiple of a → , then there is a partition of Q ∖ { 0 } into two cells, neither of which is a member of a 1 p + a 2 p + … + a m p and a member of b 1 q + b 2 q + … + b k q for any idempotents p and q in β Q d ∖ { 0 } . (Here β Q d is the Stone–Čech compactification of the set of rational numbers with the discrete topology.) In this paper we extend these results to direct sums of Q . As a corollary, we show that if b → is not a rational multiple of a → and G is any torsion free commutative group, then there do not exist idempotents p and q in β G d ∖ { 0 } such that a 1 p + a 2 p + … + a m p = b 1 q + b 2 q + … + b k q . We also show that for direct sums of finitely many copies of Q we can separate the corresponding Milliken–Taylor systems, with a similar but weaker result for the direct sum of countably many copies of Q . [ABSTRACT FROM AUTHOR]

In this paper we give a characterization of the class of completable strong (non-Archimedean) fuzzy metric spaces, in the sense of George and Veeramani. [ABSTRACT FROM AUTHOR]

For a Hausdorff space X , let PR ( X ) be the Pixley–Roy hyperspace of X . Dow and Moore [10, Proposition 2.13] noted that no compactification of PR ( 2 ω ) has countable tightness. In this paper, we will discuss when PR ( X ) has a compactification of countable tightness in general. Among other things, we will show that if a Pixley–Roy hyperspace has a compactification of countable tightness, then it will actually have a Corson compactification. [ABSTRACT FROM AUTHOR]

Approximately 10 years ago, Zambakhidze asked whether every non-zero-dimensional topological group with a bc -base is locally compact. Below we show that the small inductive dimension ind of any non-locally compact group with such a base doesn't exceed 1. We prove, however, that a σ -compact non-locally compact topological group with a bc -base is zero-dimensional. Two more results in this paper are worth mentioning: 1) if the free topological group F ( X ) of a Tychonoff space X has a bc -base, then ind ( X ) ≤ 0 , and 2) a topological group G has a bc -base if and only if G can be compactified by a zero-dimensional remainder. [ABSTRACT FROM AUTHOR]

We continue the exploration of various aspects of divisibility of ultrafilters, adding one more relation to the picture: multiplicative finite embeddability. We show that it lies between divisibility relations ∣ M and ∣ ~ . The set of its minimal elements proves to be very rich, and the ∣ ~ -hierarchy is used to get a better intuition of this richness. We find the place of the set of ∣ ~ -maximal ultrafilters among some known families of ultrafilters. Finally, we introduce new notions of largeness of subsets of N , and compare it to other such notions, important for infinite combinatorics and topological dynamics. [ABSTRACT FROM AUTHOR]