Английская Википедия:Erdős space

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Шаблон: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 <math>E\subset\ell^2</math> 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 <math>E</math> is homeomorphic to <math>E\times E</math> in the product topology. If the set of all homeomorphisms of the Euclidean space <math>\mathbb{R}^n</math> (for <math>n\ge 2</math>) that leave invariant the set <math>\mathbb{Q}^n</math> 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 <math>f(z)=e^z-1</math>. Let <math>f^n</math> denote the <math>n</math>-fold composition of <math>f</math>. The set of all points <math>z\in \mathbb C</math> such that <math>\text{Im}(f^n(z))\to\infty</math> is a collection of pairwise disjoint rays (homeomorphic copies of <math>[0,\infty)</math>), each joining an endpoint in <math>\mathbb C</math> to the point at infinity. The set of finite endpoints is homeomorphic to Erdős space <math>E</math>.

See also

• Шаблон:Annotated link

References

Шаблон:Topology-stub