Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-98-19

October 1998

Powers of Roy Spaces of Weight  aleph1

A.J. Ostaszewski


A Roy space is a zero-dimensional metric space having covering dimension at least 1. Using the technique introduced by Mrówka [4], we prove the following.

Theorem  The two known Roy spaces of weight  aleph1  (Kulesza [2] and Ostaszewski [7]) have squares of covering dimension 1.

We also show how to generalize the result to arbitrary  finite powers by indicating the changes needed in the case of cubing. Assuming the Continuum Hypothesis (CH), the corresponding theorem for the square was proved by Mrówka [4] in the case of the other then known Roy spaces, which are of weight  2aleph0.  In our case CH is unnecessary (since one does not need to map  aleph1  onto the line). There is resurgent interest in the problem of constructing Roy spaces with dimensional discrepancy larger than 1, so the current contribution presents what is hoped to be a more readable account of [4] by elucidating some of the finer details.

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