Lipscomb’s 𝐿(𝐴) space fractalized in Hilbert’s 𝑙²(𝐴) space

Source abstract

By extending adjacent-endpoint identification in Cantor’s space N ( { 0 , 1 } ) N(\{ 0,1\} ) to Baire’s space N ( A ) N(A) , we move from the unit interval I = L ( { 0 , 1 } ) I = L(\{ 0,1\} ) to L ( A ) L(A) . The metric spaces L ( A ) n + 1 L{(A)^{n + 1}} and L ( A ) ∞ L{(A)^\infty } have provided nonseparable analogues of Nöbeling’s and Urysohn’s imbedding theorems. To date, however, L ( A ) L(A) has no metric description. Here, we imbed L ( A ) L(A) in l 2 ( A ) {l^2}(A) and the induced metric yields a geometrical interpretation of L ( A ) L(A) . Except for the small last section, we are concerned with the imbedding. Once inside l 2 ( A ) {l^2}(A) , we see L ( A ) L(A) as a subspace of a "closed simplex" Δ A {\Delta ^A} having the standard basis vectors together with the origin as vertices. The part of L ( A ) L(A) in each n n -dimensional face σ n {\sigma ^n} of Δ A {\Delta ^A} is a "generalized Sierpiński Triangle" called an n n -web ω n {\omega ^n} . Topologically, ω n {\omega ^n} is L ( { 0 , 1 , … , n } ) L(\{ 0,1, \ldots ,n\} ) . For n = 2 , ω 2 n = 2,{\omega ^2} is just the usual Sierpiński Triangle in E 2 {E^2} ; for n = 3 , ω 3 n = 3,{\omega ^3} is Mandelbrot’s fractal skewed web. Thus, L ( A ) → l 2 ( A ) L(A) \to {l^2}(A) invites an extension of fractals. That is, when | A | |A| infinite, Baire’s Space N ( A ) N(A) is a "generalized code space" on | A | |A| symbols that addresses the points of the "generalized fractal" L ( A ) L(A) .