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AbstractIn this paper, we study some characterizations of q-spaces, strict q-spaces and strong q-spaces under ω-balanced topological groups. First we characterize a topological group G to be ω-balanced and a q-space whenever for each open neighborhood O of the identity in G, there is a countably compact invariant subgroup H which is of countable character in G, such that H ⊆ O and the canonical quotient mapping p : G → G/H is quasi-perfect and the quotient group G/H is metrizable. Secondly, we characterize G to be ω-balanced and a strict q-space, replacing the condition of being a q-space with an appropriate condition which is designed for the so-called “strict q-spaces”. Finally, we characterize G to be ω-balanced and a strong q-space, illustrating an additional condition to be replaced to that of q-space, or strict q-space. As applications of our three main results, we found new characterizations of the ω-narrow topological groups. [ABSTRACT FROM AUTHOR]

In this paper, the notion of a soft topological transformation group is defined and studied. For a soft topological transformation group, it is proven that a map from a soft topological space onto itself is soft homeomorphism. The collection of all soft homeomorphisms of the given soft topological space onto itself constitutes soft topological group under composition. Subsequently, it is proved that there is a homomorphism between soft topological group and the group structure on the collection of all soft homeomorphisms of given topological space. Subsequently, it is shown that the mapping space Map(Y;Y) is soft Hausdorff and verified that any subspace of the mapping space is soft Hausdorff. Additionally, it is proved that the set of all soft homeomorphisms on Y forms a soft discrete space, soft extremally disconnected space, soft Moscow space and a soft Moscow topological group. Later, it is shown that the map from a soft topological group to a mapping space is soft continuous. Finally, it is proved that distinct group structure generates distinct collection of all soft homeomorphisms of the specified soft topological space onto itself is a soft isomorphism. [ABSTRACT FROM AUTHOR]

In this paper, we introduce proximal groups that are a generalization of topological groups. A straight-forward application of proximal groups is given in the concise representation of collections of overlapping Hilbert envelope lobes attached to the peak points on oscillatory motion waveforms that occur in sequences of video frames that track object movements. [ABSTRACT FROM AUTHOR]

The paper examines various properties of D T - k -subgroup structures and addresses an open problem on the existence of topological group structures on the n -dimensional Khalimsky (K -, for brevity) topological space and Marcus–Wyse (M -, for short) topological plane. In particular, we obtain many types of totally k -disconnected or k -connected subgroups of a k -connected D T - k -group. Besides, we prove that each of the n -dimensional K -topological space and the M -topological plane cannot be a typical topological group. Unlike an existence of a D T - k -group structure of (S C k n , l , ∗) (see Proposition 4.7), we prove that neither of (S C K n , l , ∗) and (S C M l , ∗) is a topological group, where S C K n , l (respectively, S C M l ) is a simple closed K - (respectively, M -) topological curve with l elements in ℤ n (respectively, ℤ 2 ) and the operation "∗" is a special kind of binary operation for establishing a group structure of each of S C K n , l and S C M l . Finally, given a D T - k -group structure of (S C k 1 n 1 , l 1 × S C k 2 n 2 , l 2 , ∗) , we find several types of D T - k -subgroup structures of it (see Theorem 5.7). [ABSTRACT FROM AUTHOR]

This paper studies topological definitions of chain recurrence and shadowing for continuous endomorphisms of topological groups generalizing the relevant concepts for metric spaces. It is proved that in this case the sets of chain recurrent points and chain transitive component of the identity are topological subgroups. Furthermore, we show that some dynamical properties are induced by the original system on quotient spaces. These results link an algebraic property to a dynamical property. [ABSTRACT FROM AUTHOR]

Using binary function spaces, we give an example of a pseudocompact discretely selective topological group. We show that, under PFA, every compact space of countable tightness has a countable disjoint local π-base at every point. If X is a compact space of countable tightness and all non-empty open subsets of X are non-separable, then it is proved in ZFC that X possesses a countable disjoint local π-base at every point. We also establish that a Lindelöf Σ-space has the discrete shrinking property if and only if the outer π-character of any compact subset of X is uncountable. As a consequence, any non-metrizable topological group with a countable network has the discrete shrinking property. [ABSTRACT FROM AUTHOR]

We study topological groups of monotonic automorphisms on a generalized ordered space L. We find a condition that is necessary and sufficient for the set of all monotonic automorphism on L along with the function composition and the topology of point-wise convergence to be a topological group.

It is common knowledge that any topological group that satisfies the lowest separation axiom, T0, is immediately Hausdorff and completely regular; however, this is not the case for normality. This motivates us to introduce the concept of pseudo-normal groups along with pseudo-Tychonoff topological groups as generalizations of the normality and Tychonoffness of topological groups, respectively. We show that every pseudo-normal (resp. pseudo-Tychonoff) topological group is normal (resp. Tychonoff). Generally, the reverse implication of the latter does not hold. Then, we discuss their main properties in detail. To clarify these properties, we provide some examples. Finally, we establish some other results.

Using the notion of ωs-balanced semitopological group and some cardinal invariants, Kumar and Tyagi [7] characterize when a regular (T1, Hausdorff) semitopological group admits a homeomorphic embedding as a subgroup into a product of regular (T1, Hausdorff) semitopological groups with a strong development. In this paper, we show that a T0 semitopological group G is topologically isomorphic to a subgroup of a topological product of T0 semitopological groups with a strong development if and only if G is ωs-balanced. Then the main theorems of [7] are immediate consequences of our result mentioned above. [ABSTRACT FROM AUTHOR]

Under GCH, there are described the cardinalities of all Hausdorff topological groups G such that there is a nonzero Borel measure on G having the card ⁢ (G) -Suslin property and quasi-invariant with respect to an everywhere dense subgroup of G. Some connections are pointed out with the method of Kodaira and Kakutani (1950) for constructing a nonseparable translation invariant extension of the standard Lebesgue (Haar) measure on the circle group 퐒 1 . [ABSTRACT FROM AUTHOR]

Rough set theory provides valuable tools for handling and analyzing ciphertext, making it a prominent asset in cryptographic applications. Its ability to manage uncertainty and reduce complexity can enhance various aspects of ciphertext management, from pattern recognition, classification to cryptanalysis and security checks. By imposing the principles of rough sets, cryptographic systems can become more robust, efficient, and secure. The fundamental nature of the symmetric group within the context of rough topological groups makes it a powerful tool in both theoretical and applied mathematics. Some cryptographic protocols and coding theories depend on the properties of topological rough symmetric groups for security and error detection or correction. This paper aims to generalize topological rough group structures and investigate their properties. Additionally, an algorithm is established to classify the symmetric group $S_{n}$ , and experimental result is provided to explore the effectiveness of the algorithm. It provides practical tools for analyzing imprecise or incomplete data, benefiting fields such as medical diagnostics, economic forecasting, and geographical information systems.

In this paper, we introduce some concepts of ★-(quasi)-pseudometric spaces, and give an example which shows that there is a ★-quasi-pseudometric 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.

It is known that every pseudocompact topological group is precompact, we extend this result to a class of subsemigroup of topological groups. Then we use this results to prove that cancellative locally compact countably compact topological semigroups with open shifts are topological groups and to give a sufficient condition under which a locally compact monothetic topological semigroup is a compact topological group.

We completely classify the orientable infinite-type surfaces S such that PMap(S), the pure mapping class group, has automatic continuity. This classification includes surfaces with noncompact boundary. In the case of surfaces with finitely many ends and no noncompact boundary components, we prove the mapping class group Map(S) does not have automatic continuity. We also completely classify the surfaces such that PMapc(S), the subgroup of the pure mapping class group composed of elements with representatives that can be approximated by compactly supported homeomorphisms, has automatic continuity. In some cases when PMapc(S) has automatic continuity, we show any homomorphism from PMapc(S) to a countable group is trivial. [ABSTRACT FROM AUTHOR]

We study topological groups of monotonic automorphisms on a generalized ordered space L. We find a condition that is necessary and sufficient for the set of all monotonic automorphisms on L along with the function composition and the topology of point-wise convergence to be a topological group. [ABSTRACT FROM AUTHOR]

Suppose that G is a compact Hausdorff topological group and H is a closed subgroup of G. The relative commutativity degree of H in G, denoted by Pr(H;G), represents the probability that an element of H commutes with an element of G. Let D(G) be the set of all relative commutativity degrees of subgroups of G. In this paper, we will study the structure of topological groups that have exactly three relative commutativity degrees for their subgroups. In particular, we will show that for such groups, the centralizer of every non-central element is a maximal abelian subgroup. We will also provide examples of groups that have three relative commutativity degrees. [ABSTRACT FROM AUTHOR]

We prove a concentration inequality for invariant means on topological groups, namely for such adapted to a chain of amenable topological subgroups. The result is based on an application of Azuma's martingale inequality and provides a method for establishing extreme amenability. Building on this technique, we exhibit new examples of extremely amenable groups arising from von Neumann's continuous geometries. Along the way, we also answer a question by Pestov on dynamical concentration in direct products of amenable topological groups. [ABSTRACT FROM AUTHOR]

We study the separately continuous actions of semitopological monoids on pseudocompact spaces. The main aim of this paper is to generalize Lawson's results to some class of pseudocompact spaces. Also, we introduce the concept of a weak q D -space and prove that a pseudocompact space and a weak q D -space form a Grothendieck pair. As an application of the main result, we investigate the continuity of multiplication and taking inverses in subgroups of semitopological semigroups. In particular, we get that if (S , ∙) is a Tychonoff pseudocompact semitopological monoid with a quasicontinuous multiplication ∙ and G is a subgroup of S, then G is a topological group. Also, we study the continuity of operations in semitopological semilattices. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a certain topology on the group DiffF (M) of all Cr-diffeomorphisms of the foliated manifold (M; F), where r ≥ 0. This topology depends on the foliation and is called the F-compact-open topology. It coincides with the compact-open topology when F is an n-dimensional foliation. If the codimension of the foliation is n, then the convergence in this topology coincides with the pointwise convergence, where n = dim M. We prove that some subgroups of the group DiffF (M) are topological groups with the F-compact-open topology. Throughout this paper, we use smoothness of the class C∞. [ABSTRACT FROM AUTHOR]

In this paper, we examine topological partitions of topological spaces that arise in connection with continuous mappings of topological spaces. The content of the paper is closely related to such classical fundamental concepts of general topology as compactness, homogeneity, and Čech completeness. New facts related to these concepts are obtained. [ABSTRACT FROM AUTHOR]

We use Segal-Mitchison's cohomology of topological groups to define a convenient model for topological gerbes. We introduce multiplicative gerbes over topological groups in this setup and define their representations. For a specific choice of representation, we construct its category of endomorphisms, and we show that it induces a new multiplicative gerbe over another topological group. This new induced group is fiberwise Pontrjagin dual of the original one, and therefore we called the pair of multiplicative gerbes "Pontrjagin dual". We show that Pontrjagin dual multiplicative gerbes have equivalent categories of representations. In addition, we show that their monoidal centers are equivalent. Examples of Pontrjagin dual multiplicative gerbes over finite and discrete, as well as compact and non-compact, Lie groups are provided. [ABSTRACT FROM AUTHOR]

Given a sequence U = { U n : n ∈ ω } of non-empty open subsets of a space X, a family { V n : n ∈ ω } is a shrinking of U if Vn is a non-empty open set and V n ⊂ U n for every n ∈ ω . A space X has the disjoint (discrete) shrinking property if every sequence of non-empty open subsets of X has a disjoint (discrete) shrinking. We present a topological equivalent of the disjoint shrinking property in general spaces and apply it to characterize the disjoint shrinking property in topological groups. We also show that a locally convex space L has the disjoint shrinking property if and only if w (L) > ω . The space L has the discrete shrinking property if and only if L is not metrizable. We also characterize discrete selectivity in C p (X , [ 0 , 1 ]) for spaces X with a unique non-isolated point. Our results solve several published open questions. [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]

It is known that every pseudocompact topological group is precompact, we extend this result to a class of subsemigroup of topological groups. Then we use this results to prove that cancellative locally compact countably compact topological semigroups with open shifts are topological groups and to give a sufficient condition under which a locally compact mono- thetic topological semigroup is a compact topological group. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of GG-Lipschitz tracking property and GG-asymptotic tracking property in metric GG-space and obtain the equivalent conditions of GG-asymptotic tracking property in metric GG-space. In addition, we prove that the self-map ff has the GG-Lipschitz tracking property if and only if the shift map σ\sigma has the G¯\overline{G}-Lipschitz tracking property in the inverse limit space under the topological group action. These results generalize the corresponding results in [Proc. Amer. Math. Soc. 115 (1992), 573–580].

In this paper, we introduce and study a class of topologized groups called β-I-topological groups. [ABSTRACT FROM AUTHOR]

In this paper, we study the complete ⋆-metric semigroups and groups and the Raǐkov completion of invariant ⋆-metric groups. We obtain the following. (1) Let (X , d ⋆) be a complete ⋆-metric space containing a semigroup (group) G that is a dense subset of X. If the restriction of d ⋆ on G is invariant, then X can become a semigroup (group) containing G as a subgroup, and d ⋆ is invariant on X. (2) Let (G , d ⋆) be a ⋆-metric group such that d ⋆ is invariant on G. Then, (G , d ⋆) is complete if and only if (G , τ d ⋆) is Raǐkov complete. [ABSTRACT FROM AUTHOR]

In this paper, a derivative for functions f : G → H , where G is any metric divisible group and H is a metric Abelian group with a group metric, is defined. Basic differentiation theorems are stated and demonstrated. In particular, we obtain the Chain Role. [ABSTRACT FROM AUTHOR]

In this paper, we introduced several results of the topological ring. We proved that if N is a norm on the ring R, then any topology with matric space d which is defined by N is a topological ring. Also, if G is a topological group and N ≤ G, so the closure of N is also subgroup of G. Finally, we study the relation between image of f and the quotient of R\Ker (f). We showed that the ideal concept is very important to understand the ring in this study. Also, we applied this concept by new way which is topological ideal over the topological ring. [ABSTRACT FROM AUTHOR]

We prove that if a topology on the real line endows it with a topological group structure (additive) for which the interval (0 , + ∞) is an open set, so this topology is stronger than the usual topology. As a consequence we obtain characterizations of the usual topology as group topology and as ring topology. We also proved that if a topology on the real line is compatible with its usual lattice structure and is T 1 , so this topology is stronger than the usual topology, and as a consequence we obtain a characterization of the usual topology as lattice topology. [ABSTRACT FROM AUTHOR]

The orbit space of a distributive binary -space is studied. A number of its properties in the case of a compact binarily acting group are established. [ABSTRACT FROM AUTHOR]

In this paper, the algebraic structure of connected torsion-free weakly locally compact abelian groups and properties of some weakly locally compact abelian groups are established. [ABSTRACT FROM AUTHOR]

The purpose of this note is to start the systematic analysis of cofinal types of topological groups. [ABSTRACT FROM AUTHOR]

In this paper, some three-space properties of (compactly-fibered, neutrally-fibered) coset spaces are considered. Let X = G / H be a coset space and K a closed subgroup of G with H ⊂ K. It is mainly shown that (1) If G / H is neutrally-fibered, then G / H is second-countable ⇔ K / H and G / K are second-countable; (2) If G / H is compactly-fibered such that all compact (resp., countably compact) subspaces of K / H and G / K are metrizable, then all compact (resp., countably compact) subspaces of G / H are metrizable; (3) If G / H is neutrally-fibered such that K / H is second-countable and G / K an ℵ 0 -space (resp., cosmic), then G / H is an ℵ 0 -space (resp., cosmic); (4) If G / H is neutrally-fibered such that K / H is second-countable and G / K has a star-countable cs -network, then G / H has a star-countable cs -network; (5) If G / H is compactly-fibered such that K / H is locally compact metrizable and G / K stratifiable (resp., k -semi-stratifiable), then G / H is stratifiable (resp., k -semi-stratifiable). [ABSTRACT FROM AUTHOR]

This search discuss the problem between topological group and topological groupoid . We shall take some consequences of groupoid space and group space and make a relation with the tensor product, As well we discuss the action between the action of topological groupoid and the action of topological group and we attach a relation with the tensor product . Some remarks, proposition and examples are our own except if else is referred to. [ABSTRACT FROM AUTHOR]

In this note we study some topological properties of bounded sets and Bourbaki-bounded sets. Also we introduce two types of Bourbaki-bounded homomorphisms on topological groups including, n-Bourbaki-bounded homomorphisms and B-Bourbaki-bounded homomorphisms. We compare them to each other and with the class of continuous homomorphisms. So, two topologies are presented on them and we determine some properties on domain and range spaces led to Bourbaki-completeness some of these classes of homomorphisms with the given topologies. At the end of this note we focus on n-compact homomorphisms and B-compact homomorphisms briey. [ABSTRACT FROM AUTHOR]

In this paper, we determine the complex-valued solutions of the functional equation f (x σ (y)) + f (τ (y) x) = 2 f (x) f (y) for all x , y ∈ M , where M is a monoid, σ : M ⟶ M is an involutive automorphism and τ : M ⟶ M is an involutive anti-automorphism. The solutions are expressed in terms of multiplicative functions, and characters of 2-dimensional irreducible representations of M. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of p-topological group and p-irresolute topological group which are generalizations of the notion topological group. We discuss the properties of p-topological groups with illustrative examples and we provide a connected p-topological group on any group G whose cardinality is not equal to 2. Also, we prove that translations and inversion in p-topological group are phomeomorphism and demonstrate that every p-topological group is phomogenous which leads to check whether a topology on a group satisfies the conditions of p-topological group or not.

In this paper, we study the notions of point-open topology Cp(X,H), openpoint topology Ch(X,H) [resp. Ch(G,H)] and bi-point-open topology Cph(X,H) [resp. Cph(G,H)] on C(X,H) [resp.C(G,H)], the set of all continuous functions from a topological space X (topological group G) to a topological group H. In this setting, we study the countability, separation axioms and metrizability. The equivalent conditions are given so that the space Ch(G,H) is a zero-dimensional topological group. Further, if G is H⋆⋆-regular, then Ch(G,H) is Hausdorff if and only if G is discrete. It is shown that under certain conditions the topological groups Cp(X,H), Ch(X,H) and Cph(X,H) are ω-narrow. Sufficient conditions are given for the topological spaces Cp(X,H), Ch(X,H) and Cph(X,H) to be discretely selective and to have a disjoint shrinking. [ABSTRACT FROM AUTHOR]

Abstract There exists two different definitions for soft topological groups, the first due to Hida and the other due to Tariq Shah. In this paper, we give a generalization for both of them, and also, we study the topological properties for the construction of finer soft topology made by Nazmul and use the same technique to construct a soft topological group via Hida concept starting from a parametrized family of topological groups. Then, we introduce the concept of soft rough and rough soft topological group to generalize the concept of rough and soft topological group.

The notion of operator-valued frames (OVFs) by Kaftal et al. (Trans Am Math Soc 361(12):6349–6385, 2009) and G-frames by Sun (J Math Anal Appl 322(1):437–452, 2006) do not include all generalizations of notions of frames for Hilbert spaces. For this purpose, we introduce the notion of weak operator-valued frames (weak OVFs) which covers all known generalizations of frames for Hilbert spaces. Theory of weak-OVFs is more demanding than the theory of OVFs due to the following: One, the weak frame operator may not factor and another, weak frame operator may not be positive. As a first step towards a reasonable theory, we impose factorability condition on weak frame operator. We then characterize and derive dilation results. Similarity and orthogonality notions are introduced and characterized. The notion is connected with groups as well as group-like unitary systems. We also derive stability results. [ABSTRACT FROM AUTHOR]

In this paper we shall use the fundamental theorem of dynamical systems to improve some of the results from [5] concerning topological transitivity of the non-wandering set Ω(φ) and the chain recurrent set CR(φ) in compact metric spaces. [ABSTRACT FROM AUTHOR]

Given a topological group G that can be embedded as a topological subgroup into some topological vector space (over the field of reals) we say that G has invariant linear span if all linear spans of G under arbitrary embeddings into topological vector spaces are isomorphic as topological vector spaces. For an arbitrary set A let Z (A) be the direct sum of |A|-many copies of the discrete group of integers endowed with the Tychonoff product topology. We show that the topological group Z (A) has invariant linear span. This answers a question from a paper of Dikranjan et al. (J Math Anal Appl 437:1257–1282, 2016) in positive. We prove that given a non-discrete sequential space X, the free abelian topological group A(X) over X is an example of a topological group that embeds into a topological vector space but does not have invariant linear span. [ABSTRACT FROM AUTHOR]