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

 

Lipscomb's $ L(A)$ space fractalized in Hilbert's $ l\sp 2(A)$ space


Authors: S. L. Lipscomb and J. C. Perry
Journal: Proc. Amer. Math. Soc. 115 (1992), 1157-1165
MSC: Primary 54H05; Secondary 54B15, 58F08
MathSciNet review: 1093602
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Abstract: By extending adjacent-endpoint identification in Cantor's space $ N(\{ 0,1\} )$ to Baire's space $ N(A)$, we move from the unit interval $ I = L(\{ 0,1\} )$ to $ L(A)$. The metric spaces $ L{(A)^{n + 1}}$ and $ L{(A)^\infty }$ have provided nonseparable analogues of Nöbeling's and Urysohn's imbedding theorems. To date, however, $ L(A)$ has no metric description. Here, we imbed $ L(A)$ in $ {l^2}(A)$ and the induced metric yields a geometrical interpretation of $ L(A)$. Except for the small last section, we are concerned with the imbedding. Once inside $ {l^2}(A)$, we see $ L(A)$ as a subspace of a "closed simplex" $ {\Delta ^A}$ having the standard basis vectors together with the origin as vertices. The part of $ L(A)$ in each $ n$-dimensional face $ {\sigma ^n}$ of $ {\Delta ^A}$ is a "generalized Sierpiński Triangle" called an $ n$ -web $ {\omega ^n}$. Topologically, $ {\omega ^n}$ is $ L(\{ 0,1, \ldots ,n\} )$. For $ n = 2,{\omega ^2}$ is just the usual Sierpiński Triangle in $ {E^2}$; for $ n = 3,{\omega ^3}$ is Mandelbrot's fractal skewed web. Thus, $ L(A) \to {l^2}(A)$ invites an extension of fractals. That is, when $ \vert A\vert$ infinite, Baire's Space $ N(A)$ is a "generalized code space" on $ \vert A\vert$ symbols that addresses the points of the "generalized fractal" $ L(A)$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1093602-9
PII: S 0002-9939(1992)1093602-9
Keywords: Fractals, imbedding, dimension theory, metric space
Article copyright: © Copyright 1992 American Mathematical Society