# 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 the direct product . 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]

## References

- Erdős, Paul (1940), "The dimension of the rational points in Hilbert space" (PDF),
*Annals of Mathematics*, Second Series,**41**: 734–736, doi:10.2307/1968851, MR 0003191 - Dijkstra, Jan J.; van Mill, Jan (2010), "Erdős space and homeomorphism groups of manifolds",
*Memoirs of the American Mathematical Society*,**208**(979), doi:10.1090/S0065-9266-10-00579-X, ISBN 978-0-8218-4635-3, MR 2742005

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