Some Theorems on the Structure of Nearly Equicontinuous Transformation Groups

1971 ◽  
Vol 23 (3) ◽  
pp. 421-425 ◽  
Author(s):  
Fred A. Roberson
Keyword(s):  
Transformation Group ◽  
Uniform Space ◽  
Transformation Groups ◽  
Topological Spaces ◽  
Orbit Closure ◽  
Uniform Spaces ◽  
General Reference ◽  
Minimal Set ◽  
Locally Compact ◽  
Totally Disconnected

The purpose of this paper is to extend the theorems in [3; 7] to uniform spaces and to prove some additional theorems. These results are related to [4; 5]. Notation and definitions are as in the book [2]. For a general reference on nets see [6]. All topological spaces are assumed to be Hausdorff.THEOREM 1. Let (X, T, Π) be a transformation group, where X is a locally compact, locally connected, uniform space. Let E denote the set of all points at which T is equicontinuous and N = X – E. Let N be closed totally disconnected and each orbit closure in N be compact and let E be connected. Then N contains at most two minimal sets. (Note: We will assume that N ≠ ∅ so that N will contain at least one minimal set.)

scholarly journals On locally weakly almost periodic transformation groups

1982 ◽  
Vol 25 (2) ◽  
pp. 215-219
Author(s):  
Saber Elaydi
Keyword(s):  
Phase Space ◽  
Transformation Group ◽  
Transformation Groups ◽  
Almost Periodic ◽  
Locally Compact ◽  
Weakly Almost Periodic ◽  
Phase Group ◽  
Periodic Transformation

It is shown that a transformation group with a locally compact Hausdorff phase space and an abelian phase group is locally weakly almost periodic if and only if it is P-locally weakly almost periodic for some replete semigroup P in the phase group.

The Homology of Uniform Spaces

1973 ◽  
Vol 25 (3) ◽  
pp. 449-455 ◽  
Author(s):  
Mohammed Bahauddin ◽  
John Thomas
Keyword(s):  
Algebraic Topology ◽  
Uniform Space ◽  
Topological Spaces ◽  
Dimension Theory ◽  
Uniform Spaces ◽  
The Past ◽  
Body Of Knowledge ◽  
Čech Homology ◽  
Group A ◽  
Cech Cohomology

In the past thirty years, algebraic topologists have developed a great body of knowledge concerning the category of topological spaces. By contrast, corresponding problems in the category of uniform spaces have been barely touched. Lubkin [8] studied the notion of a covering space in the category of generalized uniform spaces, and suggested that much of algebraic topology could be profitably studied in this category. Deming [2] discussed the fundamental group of a generalized uniform space, and related it to the first Čech homology group. A slightly different version of Cech cohomology was defined by Kuzminov and Svedov in [7] and related to the dimension theory of uniform spaces.

scholarly journals Relations between Non-Compact Transformation Groups and Compact Transformation Groups

1971 ◽  
Vol 44 ◽  
pp. 97-117
Author(s):  
Hsin Chu
Keyword(s):  
Fundamental Group ◽  
Transformation Group ◽  
Fibre Bundle ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Transformation Groups ◽  
Almost Periodic ◽  
Minimal Set ◽  
Weakly Almost Periodic ◽  
Necessary And Sufficient

In this paper certain relations between non-compact transformation groups and compact transformation groups are studied. The notion of re-ducibility and separability of transformation groups is introduced, several necessary and sufficient conditions are established: (1) A separable transformation group to be locally weakly almost periodic, (2) A reducible and separable transformation group to be a minimal set and (3) A reducible and separable transformation group to be a fibre bundle. As applications we show, among other things, that (1) for certain reducible transformation groups its fundamental group is not trivial which is a generalization of a result in [4].

On ARU-Resolutions of Uniform Spaces

2003 ◽  
Vol 10 (2) ◽  
pp. 201-207
Author(s):  
V. Baladze
Keyword(s):  
Uniform Space ◽  
Uniform Spaces

Abstract In this paper theorems which give conditions for a uniform space to have an ARU-resolution are proved. In particular, a finitistic uniform space admits an ARU-resolution if and only if it has trivial uniform shape or it is an absolute uniform shape retract.

On Homology and Cohomology Groups of Remainders

2004 ◽  
Vol 11 (4) ◽  
pp. 613-633
Author(s):  
V. Baladze ◽  
L. Turmanidze
Keyword(s):  
Uniform Space ◽  
Cohomology Groups ◽  
Uniform Spaces ◽  
Intrinsic Characterization

Abstract Border homology and cohomology groups of pairs of uniform spaces are defined and studied. These groups give an intrinsic characterization of Čech type homology and cohomology groups of the remainder of a uniform space.

scholarly journals A study of uniformities on the space of uniformly continuous mappings

2020 ◽  
Vol 18 (1) ◽  
pp. 1478-1490
Author(s):  
Ankit Gupta ◽  
Abdulkareem Saleh Hamarsheh ◽  
Ratna Dev Sarma ◽  
Reny George
Keyword(s):  
Uniform Space ◽  
Uniform Spaces ◽  
Continuous Mappings ◽  
Uniformly Continuous

Abstract New families of uniformities are introduced on UC(X,Y) , the class of uniformly continuous mappings between X and Y, where (X,{\mathcal{U}}) and (Y,{\mathcal{V}}) are uniform spaces. Admissibility and splittingness are introduced and investigated for such uniformities. Net theory is developed to provide characterizations of admissibility and splittingness of these spaces. It is shown that the point-entourage uniform space is splitting while the entourage-entourage uniform space is admissible.

IMBEDDING OF TRANSFORMATION GROUPS IN AFFINE ONES AND RENORMALIZATION OF TWO-DIMENSIONAL HOMOGENEOUS σ-MODELS

1993 ◽  
Vol 08 (31) ◽  
pp. 2937-2942
Author(s):  
A. V. BRATCHIKOV
Keyword(s):  
Homogeneous Space ◽  
Transformation Group ◽  
Affine Group ◽  
Coupling Constants ◽  
Transformation Groups ◽  
Two Dimensional

The BLZ method for the analysis of renormalizability of the O(N)/O(N − 1) model is extended to the σ-model built on an arbitrary homogeneous space G/H and in arbitrary coordinates. For deriving Ward-Takahashi (WT) identities an imbedding of the transformation group G in an affine group is used. The structure of the renormalized action is found. All the infinities can be absorbed in a coupling constants renormalization and in a renormalization of auxiliary constants which are related to the imbedding.

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