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Erdős space

Totally disconnected topological space From Wikipedia, the free encyclopedia

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In mathematics, Erdős space is a topological space named after Paul Erdős, who described it in 1940.[1] 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.[1] The space is homeomorphic to in the product topology. If the set of all homeomorphisms of the Euclidean space (for ) that leave invariant the set of rational vectors is endowed with the compact-open topology, it becomes homeomorphic to the Erdős space.[2]

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 (homeomorphic copies of ), each joining an endpoint in to the point at infinity. The set of finite endpoints is homeomorphic to Erdős space .[3]

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