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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the dimensions of projections of compact subsets of $ {\bf R}\sp{m}$


Author: Y. Sternfeld
Journal: Proc. Amer. Math. Soc. 88 (1983), 735-742
MSC: Primary 54F45
MathSciNet review: 702310
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Abstract: In this note we continue the study of a problem considered earlier by G. Nöbling, S. Mardešić, C. Pixley and the author.

Let $ W$ be an $ n$-dimensional compact subset of $ {{\mathbf{R}}^m}$, and let $ \{ {i_1},{i_2}, \ldots ,{i_k}\} \subset \{ 1,2, \ldots ,m\} $. It is shown that if the projection $ {P_{i1,i2, \ldots ,ik}}$ satisfies a certain condition called normality on $ W$, then there exist $ {i_{k + 1}}$, $ {i_{k + 2}}, \ldots ,{i_n}$ so that $ \dim {P_{i1,i2, \ldots ,{i_k},{i_{k + 1}}, \ldots ,{i_n}}}(W) = n$. It is also shown that at least $ (_k^n)$ such normal projections do exist (for $ 1 \leqslant k \leqslant n$), and an example is constructed to show that the normality condition is not necessary.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0702310-7
PII: S 0002-9939(1983)0702310-7
Article copyright: © Copyright 1983 American Mathematical Society