Erdős space

Short description: Totally disconnected topological space

In mathematics, Erdős space is a topological space named after Paul Erdős, who described it in 1940. Erdős space is defined as a subspace [math]\displaystyle{ E\subset\ell^2 }[/math] of the Hilbert space of square summable sequences, consisting of the sequences whose elements are all rational numbers.

Erdős space is a totally disconnected, one-dimensional topological space. The space [math]\displaystyle{ E }[/math] is homeomorphic to [math]\displaystyle{ E\times E }[/math] in the product topology. If the set of all homeomorphisms of the Euclidean space [math]\displaystyle{ \mathbb{R}^n }[/math] (for [math]\displaystyle{ n\ge 2 }[/math]) that leave invariant the set [math]\displaystyle{ \mathbb{Q}^n }[/math] of rational vectors is endowed with the compact-open topology, it becomes homeomorphic to the Erdős space.

Erdős space also surfaces in complex dynamics via iteration of the function [math]\displaystyle{ f(z)=e^z-1 }[/math]. Let [math]\displaystyle{ f^n }[/math] denote the [math]\displaystyle{ n }[/math]-fold composition of [math]\displaystyle{ f }[/math]. The set of all points [math]\displaystyle{ z\in \mathbb C }[/math] such that [math]\displaystyle{ \text{Im}(f^n(z))\to\infty }[/math] is a collection of pairwise disjoint rays (homeomorphic copies of [math]\displaystyle{ [0,\infty) }[/math]), each joining an endpoint in [math]\displaystyle{ \mathbb C }[/math] to the point at infinity. The set of finite endpoints is homeomorphic to Erdős space [math]\displaystyle{ E }[/math].

See also

• List of topologies – List of concrete topologies and topological spaces