Erdős space
Encyclopedia
In mathematics
, Erdős space is a topological space
named after Paul Erdős
.
Erdős space is defined as the set E of points in the Hilbert space
l2 of square summable sequences having all coordinates rational. Erdős space is a totally disconnected, one-dimensional
topological space
. The space E is homeomorphic to the direct product
E×E. Endowed with the compact-open topology
, the set of all homeomorphisms of the Euclidean space
Rn leaving the set Qn of vectors with rational coordinates invariant is homeomorphic to the Erdős space for n ≥ 2.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Erdős space is a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
named after Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
.
Erdős space is defined as the set E of points in the Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
l2 of square summable sequences having all coordinates rational. Erdős space is a totally disconnected, one-dimensional
Dimension theory
In mathematics, dimension theory is a branch of general topology dealing with dimensional invariants of topological spaces.-See also:*Lebesgue covering dimension*Inductive dimensions *Dimension...
topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
. The space E is homeomorphic to the direct product
Direct product
In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
E×E. Endowed with the compact-open topology
Compact-open topology
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly-used topologies on function spaces, and is applied in homotopy theory and functional analysis...
, the set of all homeomorphisms of the Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn leaving the set Qn of vectors with rational coordinates invariant is homeomorphic to the Erdős space for n ≥ 2.