Evendimensionality

Evendimensionality is a concept in dimension theory describing spaces in which the dimension is uniform across the entire space, meaning every nonempty open subset has the same dimension. More precisely, a topological space X is said to have evendimensionality d if there exists an integer d ≥ 0 such that for every nonempty open set U ⊆ X, dim(U) = d, where dim denotes a specified notion of dimension (commonly the Lebesgue covering dimension). The exact value of d depends on the chosen dimension notion; in many contexts d is taken to be the covering dimension.

If X has evendimensionality, then X is said to be uniformly d-dimensional in the chosen sense. The strongest commonly studied instance occurs for manifolds and Euclidean spaces, where each nonempty open subset is homeomorphic to an open subset of R^d, yielding dim(U) = d.

Examples: The Euclidean space R^n has evendimensionality n. The sphere S^n likewise has dimension n. In contrast, a space formed by joining a line and a plane in R^3 fails to have evendimensionality since some nonempty open subsets lie within a 1-dimensional piece while others lie in a 2-dimensional piece.

Notes: Evendimensionality is a strong condition and is not automatic for all spaces; it is closely related to the idea of local constancy of the dimension function. Different dimension notions may agree on many spaces but can diverge in pathological cases. The term is used chiefly in theoretical discussions of dimension theory and in contexts where a space behaves like a d-dimensional object at all scales.

See also: covering dimension, small inductive dimension, Lebesgue dimension, dimension theory, manifolds, homogeneous spaces.