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Some Theorems on the Structure of Nearly Equicontinuous Transformation Groups

Published online by Cambridge University Press:  20 November 2018

Fred A. Roberson
Fred A. Roberson*
Affiliation:
The Florida State University, Tallahassee, Florida
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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.)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 421 - 425
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Dugundji, J., Topology (Allyn and Bacon, Inc., Boston, 1966).Google Scholar
2. Gottschalk, W. H. and Hedlund, G. A., Topological dynamics (Amer. Math. Soc. Colloquium Publication, XXXVI, Providence, Rhode Island, 1955).Google Scholar
3. Gray, W. J. and Roberson, F. A., On the near equicontinuity of transformation groups, Proc. Amer. Math. Soc. 23 (1969), 59–63.Google Scholar
4. Homma, T. and Kinoshita, S., On homeomorphism which are regular except for a finite number of points, Osaka J. Math. 7 (1955), 29–38.Google Scholar
5. Kaul, S. K., On almost regular homeomorphisms, Can. J. Math. 20 (1968), 1–6.Google Scholar
6. Kelly, J. L., General topology (D. Van Nostrand Company, Inc., Princeton, 1955).Google Scholar
7. Roberson, F. A., A theorem on near equicontinuity of transformation groups, Proc. Amer. Math. Soc. 27 (1971), 189–191.Google Scholar