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

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" $L(A)$.