Erdős 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 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 is homeomorphic to in the product topology. If the set of all homeomorphisms of the Euclidean space that leave invariant the set 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. Let denote the -fold composition of. The set of all points such that is a collection of pairwise disjoint rays, each joining an endpoint in to the point at infinity. The set of finite endpoints is homeomorphic to Erdős space.