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On the Connectedness of the Sets of Limit Points of Certain Transforms of Bounded Sequences(1)

Published online by Cambridge University Press:  20 November 2018

Dany Leviatan  and
Lee Lorch
Dany Leviatan
Affiliation:
University of Illinois, Urbana, Illinois
Lee Lorch
Affiliation:
York University, Toronto, Ontario
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The transforms discussed here are the quasi-Hausdorff (Theorem 1), the [J, ƒ(x)] (Theorem 2) and the Borel integral means (Theorem 3). We are concerned here with whether or not the limit-points of these transforms of bounded sequences form connected sets. Such a set is one which cannot be decomposed into the union of two disjoint nonempty open sets.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 14 , Issue 2 , June 1971 , pp. 175 - 181
Copyright
Copyright © Canadian Mathematical Society 1971

Footnotes

(1)

This work was assisted by a grant from the National Research Council of Canada.

References

1. Agnew, R.P., A simple sufficient condition that a method of summability be stronger than convergence, Bull. Amer. Math. Soc. 52 (1946), 128-132.Google Scholar
2. Ašić, M.D. and Adamović, D. D., Limit points of sequences in metric spaces, Amer. Math. Monthly 77 (1970), 613-616.Google Scholar
3. Barone, H.G., Limit points of sequences and their transforms by methods of summability, Duke Math. J. 5 (1939), 740-752 Google Scholar
4. Borwein, D., On a scale of Abel-type summability methods, Proc. Cambridge Philos. Soc. 53 (1957), 318-322.Google Scholar
5. Jakimovski, A., The sequence-to-function analogues to Hausdorff transformations, Bull. Res. Council Israel 8F (1960), 135-154.Google Scholar
6. Leviatan, D. and Lorch, L., A characterization of totally regular [J, f(x)] transforms, Proc. Amer. Math. Soc. 23 (1969), 315-319.Google Scholar
7. Ramanujan, M.S., On Hausdorff and quasi-Hausdorff methods of summability, Quart. J. Math., Oxford Ser. 8 (1957), 197-213.Google Scholar
8. Schaefer, Paul, Limit points of bounded sequences, Amer. Math. Monthly 75 (1968), 51.Google Scholar
9. Wells, J. H., Hausdorff transforms of bounded sequences, Proc. Amer. Math. Soc. 11 (1960), 84-86.Google Scholar