# 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 ${\displaystyle E\subset \ell ^{2}}$ 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 ${\displaystyle E}$ is homeomorphic to the direct product ${\displaystyle E\times E}$ . If the set of all homeomorphisms of the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ (for ${\displaystyle n\geq 2}$ ) that leave invariant the set ${\displaystyle \mathbb {Q} ^{n}}$ of rational vectors is endowed with the compact-open topology, it becomes homeomorphic to the Erdős space.[2]

## References

1. 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
2. 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