A Criterion for Erdős Spaces

Abstract

In 1940 Paul Erdős introduced the 'rational Hilbert space', which consists of all vectors in the real Hilbert space 2 that have only rational coordinates. He showed that this space has topological dimension one, yet it is totally disconnected and homeomorphic to its square. In this note we generalize the construction of this peculiar space and we consider all subspaces E of the Banach spaces p that are constructed as 'products' of zero-dimensional subsets En of R. We present an easily applied criterion for deciding whether a general space of this type is one dimensional. As an application we find that if such an E is closed in p , then it is homeomorphic to complete Erdős space if and only if dim E > 0 and every En is zero dimensional. The topology on p is generated by the norm LetR be the compactification [−∞, ∞] of R. We extend the p-norm overRN by putting z = ∞ for each z ∈R N \ p . For the remainder of this note let E 1 , E 2 , . . . be a fixed sequence of subsets of R and let E = {z ∈ p : z n ∈ E n for every n ∈ N} be a corresponding subspace of some fixed p . If we choose p = 2 and E n = Q for every n, then E is called Erdős space E and, if E n = R \ Q, then we obtain complete Erdős space E c (cf. [9] and [10]). Erdős [9] proved that Erdős space and complete Erdős space have topological dimension one. We present an easily applied criterion for deciding whether a general space of the type E is one dimensional. As an application we find that if E is closed in p , then it is homeomorphic to complete Erdős space if and only if dim E > 0 and every E n is zero dimensional. Other applications are particularly simple models of complete Erdős space and the Lelek fan. 595

Characterizing stable complete Erdős space

“…According to Kawamura, Oversteegen and Tymchatyn [19] complete Erdős space is homeomorphic to G ϕ 0 ; see also Dijkstra [7]. Define the function ψ :…”

“…This follows from Erdős' proof [18] that these spaces have the property that every clopen nonempty subset has diameter at least 1/2 with respect to the Hilbert norm; see also Dijkstra [7]. A cohesive space is obviously at least one-dimensional at every point but the converse is not valid; see Dijkstra [8].…”

“…Proof. We begin by presenting a particularly elegant model of E c that is featured in Dijkstra [7] and called harmonic Erdős space. Consider the Cantor set C = 2 ω = {0, 1} ω with its standard boolean group structure .…”

“…It is shown in [7] that (E h , d) is homeomorphic to E c and that d is complete. Consider now the space E ω h = i∈ω E h with the function ψ : E ω h → [0, 1] that is given by (6.5) ψ(x) = max i∈ω (min{1/(i + 1), ϕ(x i )}), where x = (x 0 , x 1 , .…”

“…c , (5) X admits a closed imbedding in E ω c , (6) X is homeomorphic to a G δ -subset of E ω c , (7) there is an ω-USC function χ : C → I ω such that C is complete, dim C = 0, and X ≈ G χ 0 , (8) there is an ω-USC function χ : C → I ω such that C is compact, dim C = 0, and X ≈ G χ 0 , and (9) X is almost zero-dimensional and complete.…”