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.^[1] The space ${\displaystyle E}$ is homeomorphic to ${\displaystyle E\times E}$ in the product topology. 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]

Erdős space also surfaces in complex dynamics via iteration of the function ${\displaystyle f(z)=e^{z}-1}$. Let ${\displaystyle f^{n}}$ denote the ${\displaystyle n}$-fold composition of ${\displaystyle f}$. The set of all points ${\displaystyle z\in \mathbb {C} }$ such that ${\displaystyle {\text{Im}}(f^{n}(z))\to \infty }$ is a collection of pairwise disjoint rays (homeomorphic copies of ${\displaystyle [0,\infty )}$), each joining an endpoint in ${\displaystyle \mathbb {C} }$ to the point at infinity. The set of finite endpoints is homeomorphic to Erdős space ${\displaystyle E}$.^[3]

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