Erdős space

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. 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 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 (homeomorphic copies of ) in the complex plane. The set of all finite endpoints of these rays is homeomorphic to .[3] This representation can also be described as the set of all points such that (a) the iterates of escape to in the imaginary direction, and (b) is accessible via a continuous curve of points whose iterates attract to .

See alsoEdit


  1. ^ Erdős, Paul (1940), "The dimension of the rational points in Hilbert space" (PDF), Annals of Mathematics, Second Series, 41 (4): 734–736, doi:10.2307/1968851, JSTOR 1968851, MR 0003191
  2. ^ Dijkstra, Jan J.; van Mill, Jan (2010), "Erdős space and homeomorphism groups of manifolds" (PDF), Memoirs of the American Mathematical Society, 208 (979), doi:10.1090/S0065-9266-10-00579-X, ISBN 978-0-8218-4635-3, MR 2742005
  3. ^ Lipham, David S. (2020-05-09). "Erdős space in Julia sets". arXiv:2004.12976 [math.DS].