## Lipscomb’s $L(A)$ space fractalized in Hilbert’s $l^ 2(A)$ space

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- by S. L. Lipscomb and J. C. Perry PDF
- Proc. Amer. Math. Soc.
**115**(1992), 1157-1165 Request permission

## 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 $|A|$ infinite, Baire’s Space $N(A)$ is a "generalized code space" on $|A|$ symbols that addresses the points of the "generalized fractal" $L(A)$.

## References

- Michael Barnsley,
*Fractals everywhere*, Academic Press, Inc., Boston, MA, 1988. MR**977274** - James Dugundji,
*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606** - Hans-Joachim Kowalsky,
*Einbettung metrischer Räume*, Arch. Math. (Basel)**8**(1957), 336–339 (German). MR**91451**, DOI 10.1007/BF01900142
S. L. Lipscomb, - Stephen Leon Lipscomb,
*A universal one-dimensional metric space*, TOPO 72—general topology and its applications (Proc. Second Pittsburgh Internat. Conf., Pittsburgh, Pa., 1972; dedicated to the memory of Johannes H. de Groot), Lecture Notes in Math., Vol. 378, Springer, Berlin, 1974, pp. 248–257. MR**0358738** - Stephen Leon Lipscomb,
*On imbedding finite-dimensional metric spaces*, Trans. Amer. Math. Soc.**211**(1975), 143–160. MR**380751**, DOI 10.1090/S0002-9947-1975-0380751-8 - Stephen Leon Lipscomb,
*An imbedding theorem for metric spaces*, Proc. Amer. Math. Soc.**55**(1976), no. 1, 165–169. MR**391034**, DOI 10.1090/S0002-9939-1976-0391034-0 - Benoit B. Mandelbrot,
*The fractal geometry of nature*, Schriftenreihe für den Referenten. [Series for the Referee], W. H. Freeman and Co., San Francisco, Calif., 1982. MR**665254** - Kiiti Morita and Sitiro Hanai,
*Closed mappings and metric spaces*, Proc. Japan Acad.**32**(1956), 10–14. MR**87077** - James R. Munkres,
*Elements of algebraic topology*, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. MR**755006** - J. Nagata,
*Topics in dimension theory*, General topology and its relations to modern analysis and algebra, V (Prague, 1981) Sigma Ser. Pure Math., vol. 3, Heldermann, Berlin, 1983, pp. 497–506. MR**698449**
—, - Jun-iti Nagata,
*A remark on general imbedding theorems in dimension theory*, Proc. Japan Acad.**39**(1963), 197–199. MR**164319**
G. Nöbeling, - A. H. Stone,
*Metrizability of decomposition spaces*, Proc. Amer. Math. Soc.**7**(1956), 690–700. MR**87078**, DOI 10.1090/S0002-9939-1956-0087078-6 - Paul Urysohn,
*Zum Metrisationsproblem*, Math. Ann.**94**(1925), no. 1, 309–315 (German). MR**1512260**, DOI 10.1007/BF01208661

*Imbedding one dimensional metric spaces*, Univ. of Virginia Dissertation, University Microfilms, Ann Arbor, MI, 1973.

*A survey of dimension theory*. III, Proc. Steklov Inst. Math.

**4**(1984), 201-213.

*Über eine*$n$

*-dimensionale Universalmenge im*${R_{2n + 1}}$, Math. Ann.

**104**(1930), 71-80.

## Additional Information

- © Copyright 1992 American Mathematical Society
- 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