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

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Topological spaces