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Discontinuity Conditions on Transformation Groups

Published online by Cambridge University Press:  20 November 2018

D. V. Thompson
D. V. Thompson*
Affiliation:
University of Saskatchewan, Regina, Saskatchewan
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Throughout this paper, (X, T, π) is a topological transformation group [1], L={x∊X:xt=x for some t∊{e}} and 0=X—L is nonempty; standard topological concepts are used as defined in [2].

The problem to be considered here has been studied in [3] and [6]. In [3], X is assumed to be a compact metric space, and each t e T satisfies a convergence condition on certain subsets of X. Under these conditions, Kaul proved that if T is equicontinuous on 0, then the group properties of discontinuity, proper discontinuity, and Sperner's condition (see Definition 1) are equivalent.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 3 , September 1972 , pp. 417 - 419
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Gottschalk, W. H. and Hedlund, G. A., Topological Dynamics, Colloq. Publ., Amer. Math. Soc, 1955.Google Scholar
2. Kelley, J. L., General topology, Van Nostrand, Princeton, N.J., 1955.Google Scholar
3. Kaul, S. K., On a transformation group, Canad. J. Math. 21 (1969), 935-941.Google Scholar
4. Kaul, S. K., Compact subsets in function spaces, Canad. Math. Bull. 12 (1969), 461-466.Google Scholar
5. Kaul, S. K., On the irregular sets of a transformation group, (to appear).Google Scholar
6. Kinoshita, S., Notes on discontinuous transformation groups, (unpublished).Google Scholar