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 ⊂ ℓ 2 {\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 E {\displaystyle E} is homeomorphic to E × E {\displaystyle E\times E} in the product topology. If the set of all homeomorphisms of the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} (for n ≥ 2 {\displaystyle n\geq 2} ) that leave invariant the set Q n {\displaystyle \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 {\displaystyle f(z)=e^{z}-1} . Let f n {\displaystyle f^{n}} denote the n {\displaystyle n} -fold composition of f {\displaystyle f} . The set of all points z ∈ C {\displaystyle z\in \mathbb {C} } such that Im ( f n ( z ) ) → ∞ {\displaystyle {\text{Im}}(f^{n}(z))\to \infty } is a collection of pairwise disjoint rays (homeomorphic copies of [ 0 , ∞ ) {\displaystyle [0,\infty )} ), each joining an endpoint in C {\displaystyle \mathbb {C} } to the point at infinity. The set of finite endpoints is homeomorphic to Erdős space E {\displaystyle E} .
See also List of topologies – List of concrete topologies and topological spaces
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