On the topology of generalized quotients (original) (raw)

Generalized quotient topologies

Acta Mathematica Hungarica, 2010

A definition of a generalized quotient topology is given and some characterizations of this concept, up to generalized homeomorphisms, are furnished. For the first approach, we exhibit a monotonic map spanning that generalized quotient topology. We also prove that the notions of generalized normality and generalized compactness are preserved by those quotient structures.

Quotient spaces with strong subgyrogroups

Cornell University - arXiv, 2022

In this paper, we mainly investigate the quotient spaces G/H when G is a strongly topological gyrogroup and H is a strong subgyrogroup of G. It is shown that if G is a strongly topological gyrogroup, H is a closed strong subgyrogroup of G and H is inner neutral, then the quotient space G/H is first-countable if and only if G/H is a bisequential space if and only if G/H is a weakly first-countable space if and only if G/H is a csf-countable and sequential α 7-space. Moreover, it is shown that if G is a strongly topological gyrogroup and H is a locally compact strong subgyrogroup of G, then there exists an open neighborhood U of the identity element 0 such that π(U) is closed in G/H and the restriction of π to U is a perfect mapping from U onto the subspace π(U); if H is a locally compact metrizable strong subgyrogroup of G and the quotient space G/H is sequential, then G is also sequential; if H is a closed first-countable and separable strong subgyrogroup of G, the quotient space G/H is an ℵ 0-space, then G is an ℵ 0-space; if the quotient space G/H is a cosmic space, then G is also a cosmic space; if the quotient space G/H has a star-countable cs-network or star-countable wcs *-network, then G also has a star-countable cs-network or star-countable wcs *-network, respectively.

Some Study on the Topological Structure on Semigroups

WSEAS TRANSACTIONS ON MATHEMATICS

Some studies related to the topological structure of semigroups are provided. In, [3], considering and investigating the properties of the collection A of all the proper uniformly strongly prime ideals of a Γ - semigroup S, such study starts by constructing a topology τA on A using a closure operator defined in terms of the intersection and inclusion relation among these ideals of Γ-semigroup S, which is a generalization of the semigroup. In this paper, we introduce three other classes of ideals in semigroups called maximal ideals, prime ideals and strongly irreducible ideals, respectively. Investigating properties of the collection M, B and S of all proper maximal ideals, prime ideals and strongly irreducible ideals, respectively, of a semigroup S, we construct the respective topologies on them. The respective obtained topological spaces are called the structure spaces of the semigroup S. We study several principal topological axioms and properties in those structure spaces of semi...

The separable quotient problem for topological groups

Israel Journal of Mathematics

The famous Banach-Mazur problem, which asks if every infinitedimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Fréchet spaces. In this paper the analogous problem for topological groups is investigated. Indeed there are four natural analogues: Does every non-totally disconnected topological group have a separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. All four questions are answered here in the negative. However, positive answers are proved for important classes of topological groups including (a) all compact groups; (b) all locally compact abelian groups; (c) all σ-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all σ-compact pro-Lie groups; (f) all pseudocompact groups. Negative answers are proved for precompact groups.

A class of quotient spaces in strongly topological gyrogroups

2021

Quotient space is a class of the most important topological spaces in the research of topology. In this paper, we show that if (G, τ,⊕) is a strongly topological gyrogroup with a symmetric neighborhood base U at 0 and H is an admissible subgyrogroup generated from U , then G/H is first-countable if and only if it is metrizable. Moreover, if H is neutral and G/H is Fréchet-Urysohn with an ω-base, then G/H is first-countable. Therefore, we obtain that if H is neutral, then G/H is metrizable if and only if G/H is Fréchet-Urysohn with an ω-base. Finally, it is shown that if H is neutral, πχ(G/H) = χ(G/H) and πω(G/H) = ω(G/H).

Semicontinuous groups and separation properties

International Journal of Mathematics and Mathematical Sciences, 1992

In 1948, Samuel [2] pointed out that the intersection of two group topologies need not be a group topology. However, a number of properties that hold for a group topology still hold for a topological space that is an intersection of group topologies. In order to study these properties, we shall describe a class of topologies that can be placed on a group which we call semicontinuous topologies. (We point out here that Fuchs [1] calls these spaces semitopological groups). One important attribute of topological groups is separation. In particular, a topological group is Hansdorff if and only if the identity is a closed subset. While this is not true for semicontinuous groups, we shall see that an interesting "echo" of this property is true. For each group G we have a bijection inv: G-,G defined by inv (x) x-1. Also for any fixed a E G we have bijections la:G-,G defined by la(x) az and ra:G-G defined by

On semi-g-regular and semi-g-normal spaces

Demonstratio Mathematica

The aim of this paper is to introduce and study two new classes of spaces, called semi-g-regular and semi-g-normal spaces. Semi-g-regularity and semi-g-normality are separation properties obtained by utilizing semi-generalized closed sets. Recall that a subset A of a topological space (X, τ ) is called semi-generalized closed, briefly sg-closed, if the semi-closure of A ⊆ X is a subset of U ⊆ X whenever A is a subset of U and U is semi-open in (X, τ ) . * 2000 Math. Subject Classification -Primary: 54A05 ; Secondary: 54D10.

Certain quotient spaces are countably separated, III

Journal of Functional Analysis, 1976

Let G be a locally compact group, N a closed normal subgroup, and LY a multiplier for G such that every a-representation of G is Type I and such that every or-representation of N is Type I. Let i?a be the a-dual space of N. Necessary and sufficient conditions are given such that Ra/G is countably separated. Generalizations of many known sufficient conditions for countable separability are easy corollaries.

Semigroups of left quotients—the uniqueness problem

Proceedings of the Edinburgh Mathematical Society, 1992

Let S be a subsemigroup of a semigroup Q. Then Q is a semigroup of left quotients of S if every element of Q can be written as a*b, where a lies in a group -class of Q and a* is the inverse of a in this group; in addition, we insist that every element of S satisfying a weak cancellation condition named square-cancellable lie in a subgroup of Q.J. B. Fountain and M. Petrich gave an example of a semigroup having two non-isomorphic semigroups of left quotients. More positive results are available if we restrict the classes of semigroups from which the semigroups of left quotients may come. For example, a semigroup has at most one bisimple inverse ω-semigroup of left quotients. The crux of the matter is the restrictions to a semigroup S of Green's relations ℛ and ℒ in a semigroup of quotients of S. With this in mind we give necessary and sufficient conditions for two semigroups of left quotients of S to be isomorphic under an isomorphism fixing S pointwise.The above result is then u...

Convergence semigroup actions: generalized quotients

Applied General Topology, 2009

Continuous actions of a convergence semigroup are investigated in the category of convergence spaces. Invariance properties of actions as well as properties of a generalized quotient space are presented 2000 AMS Classification: 54A20, 54B15.