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 emerges in complex dynamics. Let be the complex exponential mapping defined by. Let denote the -fold composition of. Then the set of all points such that as forms a collection of pairwise disjoint rays in the complex plane. The set of all finite endpoints of these rays is homeomorphic to. This representation can also be described as the set of all points such that the iterates of escape to in the imaginary direction, and is accessible via a continuous curve of points whose iterates attract to.
