Erdős space

Template:Short description In mathematics, Erdős space is a topological space named after Paul Erdős, who described it in 1940. Erdős space is defined as a subspace $E\subset {\ell }^2$ 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 $E$ is homeomorphic to $E\times E$ in the product topology. If the set of all homeomorphisms of the Euclidean space ${ℝ}^n$ (for $n\ge 2$) that leave invariant the set ${\mathbb{Q}}^n$ 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 $f(z)=e^z-1$. Let $f^n$ denote the $n$-fold composition of $f$. The set of all points $z\in ℂ$ such that $\text{Im}(f^n(z))\to \mathrm{\infty }$ is a collection of pairwise disjoint rays (homeomorphic copies of $[0,\mathrm{\infty })$), each joining an endpoint in $ℂ$ to the point at infinity. The set of finite endpoints is homeomorphic to Erdős space $E$.

See also

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References

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