Topologies invariant under a group action

Proceedings of the American Mathematical Society, 1961

Baghdad Science Journal

In this paper, we define some generalizations of topological group namely -topological group, -topological group and -topological group with illustrative examples. Also, we define grill topological group with respect to a grill. Later, we deliberate the quotient on generalizations of topological group in particular -topological group. Moreover, we model a robotic system which relays on the quotient of -topological group.

An ideal on a set X is a nonempty collection of subsets of X which sat- isfies the following conditions (1)A ∈ I and B ⊂ A implies B ∈ I; (2) A ∈ I and B ∈ I implies A ∪ B ∈ I. Given a topological space (X,  ) an ideal I on X and A ⊂ X, ℜa(A) is defined as ∪{U ∈  a : U−A ∈ I}, where the family of all a-open sets of X forms a topology [5, 6], denoted by  a. A topology, denoted  a  , finer than  a is generated by the basis (I,  ) = {V − I : V ∈  a(x), I ∈ I}, and a topology, denoted hℜa( )i coarser than  a is generated by the basis ℜa( ) = {ℜa(U) : U ∈  a}. In this paper A bijection f : (X, , I) → (X, , J) is called a A∗- homeomorphism if f : (X,  a  ) → (Y, a  ) is a homeomorphism, ℜa- homeomorphism if f : (X,ℜa( )) → (Y,ℜa()) is a homeomorphism. Properties preserved by A∗-homeomorphism are studied as well as nec- essary and sufficient conditions for a ℜa-homeomorphism to be a A∗- homeomorphism.

Open Problems in Topology II, 2007

2010

Let X be a Hausdorff topological group and G a locally compact subgroup of X. We show that the natural action of G on X is proper in the sense of R. Palais. This is applied to prove that there exists a closed set F ⊂ X such that FG = X and the restriction of the quotient projection X → X/G to F is a perfect map F → X/G. This is a key result to prove that many topological properties (among them, paracompactness and normality) are transferred from X to X/G, and some others are transferred from X/G to X. Yet another application leads to the inequality dimX ≤ dimX/G+dimG for every paracompact topological group X and a locally compact subgroup G of X having a compact group of connected components.

Pacific Journal of Mathematics, 1982

This document is describing and proving a collection of propositions concerning topological spaces and topological groups. In particular, it shows that if G is a topological group, then letting H be the closure of {1}, we show that H is a closed normal subgroup of G and G/H is a T_{3 1/2} topological group.

2015

In this note, for a topological group , we introduce a new concept as bounded topological group, that is, is called bounded, if for every neighborhood of identity element of , there is a natural number such that . We study some properties of this new concept and its relationships with other topological properties of topological groups.

Bulletin of The Australian Mathematical Society, 1969

It has been shown by 0. Stephen that the number N of open sets in a non-discrete topology on a finite set with n elements is not greater than 3 x 2 . We show that for admissable n/r topologies on a finite group N < 2 , where r is the least order of its non-trivial normal subgroups. This is clearly a sharper bound.

Transactions of the American Mathematical Society, 1967

The investigations leading to this paper were suggested by the papers of Hewitt [3] and Ross [5]. In [3] Hewitt was interested in proving that if an abelian group is locally compact in two topologies, one strictly stronger than the other, there is a character continuous in one topology and discontinuous in the other (actually a special case of a theorem of Kaplansky-see Theorem 1.1 of [2]). Actually Hewitt proved a stronger result. His arguments were based on the fact that both the additive group of reals, and the multiplicative group of complex numbers of absolute value 1 have the property that every stronger locally compact group topology is discrete. A natural question to ask is what other groups have this property. The answer is very simple (2.1 of this paper). We consider in the second section of this paper the obvious generalization. Namely which groups have the property that there are only finitely many stronger locally compact group topologies. The investigations in the first section of this paper were suggested by the paper of Ross [5]. Ross was considering the same question as Hewitt, and was led to consider the relationship between two locally compact group topologies on a group G such that G has the same closed subgroups in the two topologies. We investigate this further in the first section of this paper, and are able to say that many of the properties of O as a topological group can be recovered once we know the closed subgroups. 1. Closed subgroups of locally compact groups. We consider here what we can say about a locally compact group G once we know its closed subgroups. Throughout this section of the paper R will denote the additive group of real numbers, T will denote the multiplicative group of complex numbers of absolute value 1, and, except for 7? all our groups will be written multiplicatively. 1.1. Theorem. The closed subgroups of G determine the open subgroups ofG. Proof. In case G is abelian, the proof is very easy, for then a subgroup is open if and only if every subgroup containing it is closed. In the nonabelian case this does not characterize the open subgroups (every subgroup of 57(3, C) containing SU{3) is closed, and in fact is either SU{3) or SL{3, C), but SU{3) is not open). Thus for the general case we proceed by the following steps. (1) The closed subgroups of G determine the identity component C70 of G. In fact G0 is the intersection of the closed subgroups 77 with the property that for