X-Nico

unusual facts about metric space



Exterior space

Shape theory: Many shape invariants (Borsuk groups, Quigley inward and approaching groups) of a compact metric space can be obtained as exterior homotopy groups of the exterior space determined by the open neighborhoods of a compact metric space embedded in the Hilbert cube.

Vitali covering lemma

Finite version: Let B {1}, \ldots, B {n} be any finite collection of balls contained in d-dimensional Euclidean space Rd (or, more generally, in an arbitrary metric space).


see also

Metric dimension

Equilateral dimension of a metric space (also called the metric dimension), the maximum number of points at equal distances from each other

Hausdorff dimension, an extended non-negative real number associated with any metric space that generalizes the notion of the dimension of a real vector space