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⊂ℓ2{\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 Rn{\displaystyle \mathbb {R} ^{n}} (for n≥2{\displaystyle n\geq 2}) that...

In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) are connected; in a totally disconnected space, these are the only connected subsets. An important example of a totally disconnected space is the Cantor set, which is homeomorphic to the set of p-adic integers. Another example, playing a key role in algebraic number theory, is the field Qp of p-adic numbers. == Definition == A topological space X{\...