The Independence of p of the Lipscomb's L(A) Space Fractalized in l^{p}(A)

TL;DR

This paper demonstrates that the topological structure of Lipscomb's space embedded in l^p(A) is independent of p, showing that the attractors for different p values are actually the same space.

Contribution

It proves that the Lipscomb's space attractor in l^p(A) is topologically independent of p, extending previous results and providing a detailed description of convergent sequences.

Findings

The attractor {}_p^A is equal for all p,q in [1, )

The topological structure of {}_p^A does not depend on p

Convergent sequences in {}_p^A are fully characterized

Abstract

In one of our previous papers we proved that, for an infinite set A and p\in[1,\infty), the embedded version of the Lipscomb's space L(A) in l^{p}(A), p\in[1,\infty), with the metric induced from l^{p}(A), denoted by {\omega}_{p}^{A}, is the attractor of an infinite iterated function system comprising affine transformations of l^{p}(A). In the present paper we point out that {\omega}_{p}^{A}={\omega}_{q}^{A}, for all p,q\in[1,\infty) and, by providing a complete description of the convergent sequences from {\omega}_{p}^{A}, we prove that the topological structure of {\omega}_{p}^{A} is independent of p.