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Finite powers of strong measure zero sets

Published online by Cambridge University Press:  12 March 2014

Marion Scheepers
Marion Scheepers*
Affiliation:
Department of Mathematics, Boise State University, 1910 University Drive Boise, Idaho 8372 5, USA E-mail: mscheep@micron.net
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Abstract

In a previous paper—[17]—we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove:

1. If a metric space has a covering property of Hurewicz and has strong measure zero, then its product with any strong measure zero metric space is a strong measure zero metric space (Theorem 1 and Lemma 3).

2. A subspace X of a σ-compact metric space Y has strong measure zero if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 9).

3. A subspace X of a σ-compact metric space Y has strong measure zero in all finite powers if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 12).

Then 2 and 3 yield characterizations of strong measure zeroness for σ-totally bounded metric spaces in terms of Ramseyan theorems.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 3 , September 1999 , pp. 1295 - 1306
Copyright
Copyright © Association for Symbolic Logic 1999

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References

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