The Topology of a subset of l_p as p is Varied

摘要

For 1 <= p < q < infinity and X is contained in l_p is contained in l_q, the topology on X viewed as a subset of l_p may differ from that on X viewed as a subset of l_q, We investigate this difference. With regard to dimension, we improve results in [9] by showing the possible values for dim_q X - dim_p X are exactly the integers >= -1, and that if dim_p X not= dim_q X, then X is not contained in l_m for any m < P. The key example is the set K(p) which is homeomorphic to the irrationals when viewed as a subset of l_p but, for all q > p, K(p) is homeomorphic to the Hilbert cube viewed as a subset of l_q. Additional examples, essentially closed versions of the well known Erdos example [7] all of which are weakly closed line free subgroups of l_p, demonstrate the dimension properties of such spaces. Theorem 3.2 states that if 1 <= p < q < r <= infinity, then X is bounded in l_p, implies (X, d_q) and (X, d_r) are homeomorphic via the identity map. This theorem, which is used in the above, has an application to linear spaces which is discussed in Section 6. Despite this theorem, in Section 4 we are able to give an example of countable subset Z of l_1 such that none of (Z, d_p), (Z, d_p), and (Z, d_r) are homeomorphic.