Cauchy action on filter spaces

N. Rath


A Cauchy group (G,D,·) has a Cauchy-action on a filter space (X,C), if it acts in a compatible manner. A new filter-based method is proposed in this paper for the notion of group-action, from which the properties of this action such as transitiveness and its compatibility with various modifications of the G-space (X,C) are determined. There is a close link between the Cauchy action and the induced continuous action on the underlying G-space, which is explored here. In addition, a possible extension of a Cauchy-action to the completion of the underlying G-space is discussed. These new results confirm and generalize some of the properties of group action in a topological context.


continuous action; Cauchy map; G-space; filter space and its modifications; completions

Subject classification

54A20; 54B05; 54C20; 54C05.

Full Text:



R. Beattie and H.-P. Butzman, Convergence Structures and Applications to Functional Analysis, Kluwer Academic Publ., Netherlands, 2002.

H. Boustique, P. Mikusinki and G. Richardson, Convergence semigroup actions, Applied General Topology 10 (2009), 173-186.

A. M. Carstens and D. C. Kent, A note on products of convergence spaces, Math. Ann. 182 (1969,) 40-44.

A. Császár, $lambda λ-complete Filter Spaces, Acta. Math. Hungar. 70 (1996), 75-87.

E. Colebunders, H. Boustique, P. Mikusiski and G. Richardson, Convergence Approach spaces: Actions, Applied General Topology 24 (2009), 147-161.

J. D. Dixon and B. Mortimer, Permutation groups, Springer-Verlag, New York, 1996.

R. Friuc and D. C. Kent, Completion functors for Cauchy groups, Internat. Jour. Math. and Math. Sci. (1981), 55-65.

H. H. Keller, Die Limes-Uniformisierbarkeit der Limesraüme, Math. Ann. 176 (1968), 334-341.

D. C. Kent and R. R. de Eguino, On products of Cauchy completions, Math. Nachr. 155 (1992), 47-55.

D. C. Kent and N. Rath, Filter spaces, Applied Categorical Structures 1 (1993), 297-309.

D. C. Kent and N. Rath, On completions of filter spaces, Annals of the New York Academy of Sciences 767 (1995), 97-107.

G. Minkler, J. Minkler and G. Richardson, Extensions for filter spaces, Acta. Math. Hungar. 82 (1999), 301-310.

V. Pestov, Topological groups: where to from here, Topology Proceedings 24 (1999), 421-502.

N. C. Phillips, Equivariant K-theory and freeness of group actions on C*-algebras, Lecture Notes in Mathematics, Springer, New York, 2006.

G. Preuss, Improvement of Cauchy spaces, Questions Answ. General Topology 9 (1991), 159-166.

N. Rath, Action of convergence groups, Topology Proceedings 27 (2003), 601-612.

N. Rath, Completions of filter semigroup, Acta. Math. Hungar. 107 (2005), 45-54.

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