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