Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1992-1093602-9
MathSciNet review: 1093602
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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)$.


References [Enhancements On Off] (What's this?)

  • [1] M. Barnsley, Fractals everywhere, Academic Press, Boston, MA, 1988. MR 977274 (90e:58080)
  • [2] J. Dungundji, Topology, Allyn and Bacon, Boston, MA, 1966. MR 0193606 (33:1824)
  • [3] H. J. Kowalsky, Einbettung metrischea räume, Arch. Math. 8 (1957), 336-339. MR 0091451 (19:971e)
  • [4] S. L. Lipscomb, Imbedding one dimensional metric spaces, Univ. of Virginia Dissertation, University Microfilms, Ann Arbor, MI, 1973.
  • [5] -, A universal one-dimensional metric space, TOPO 72 General Topology and Its Appl., Lecture Notes in Math., vol. 378, Springer-Verlag, Berlin, Heidelberg, New York, pp. 248-257. MR 0358738 (50:11197)
  • [6] -, On imbedding finite-dimensional metric spaces, Trans. Amer. Math. Soc. 211 (1975), 143-160. MR 0380751 (52:1648)
  • [7] -, An imbedding theorem for metric spaces, Proc. Amer. Math. Soc. 55 (1976), 165-169. MR 0391034 (52:11856)
  • [8] B. B. Mandelbrot, The fractal geometry of nature, W. H. Freeman, New York, 1983. MR 665254 (84h:00021)
  • [9] K. Morita and S. Hanna, Closed mappings and metric spaces, Proc. Japan Acad. Sci. 32 (1956), 10-14. MR 0087077 (19:299a)
  • [10] J. R. Munkres, Elements of algebraic topology, Benjamin/Cummings, Reading, MA, 1984. MR 755006 (85m:55001)
  • [11] J. Nagata, Modern dimension theory, Vol. 2 (Sigma Series in Pure Mathematics) Heldermann Verlag, Berlin, 1983. MR 698449 (84d:54062)
  • [12] -, A survey of dimension theory. III, Proc. Steklov Inst. Math. 4 (1984), 201-213.
  • [13] -, A remark on general imbedding theorems in dimension theory, Proc. Japan Acad. Sci. 39 (1963), 197-199. MR 0164319 (29:1616)
  • [14] G. Nöbeling, Über eine $ n$-dimensionale Universalmenge im $ {R_{2n + 1}}$, Math. Ann. 104 (1930), 71-80.
  • [15] A. H. Stone, Metrizability of decomposition spaces, Proc. Amer. Math. Soc. 7 (1956), 690-700. MR 0087078 (19:299b)
  • [16] P. Urysohn, Zum metrisationsproblem, Math. Ann. 94 (1925), 309-315. MR 1512260

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54H05, 54B15, 58F08

Retrieve articles in all journals with MSC: 54H05, 54B15, 58F08


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1093602-9
Keywords: Fractals, imbedding, dimension theory, metric space
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society