LebesgueAbdeckungsdimension

LebesgueAbdeckungsdimension, commonly referred to as Lebesgue covering dimension, is a topological invariant used to characterize the dimensionality of a topological space. For a given space X, the Lebesgue covering dimension, denoted dim X, is the smallest non‑negative integer n such that every open cover of X has an open refinement in which no point is contained in more than n + 1 elements. If no such integer exists, the dimension is defined to be infinite. In general, the definition applies to arbitrary topological spaces, but the most frequent applications are for separable metric spaces, where the concept behaves well.

For Euclidean spaces ℝ^n the Lebesgue covering dimension equals the usual integer n; thus ℝ^2 has covering dimension 2, while ℝ^0 is zero‑dimensional. The dimension of a closed interval and of a cube also matches their Euclidean dimension. More generally, compact metric spaces with finite covering dimension have a strong link to the small inductive dimension, which coincides with the Lebesgue dimension for metrizable spaces that are separable and finite‑dimensional.

Important properties of the covering dimension include monotonicity—any subspace inherits a dimension no larger than that of the ambient space—and its invariance under homeomorphism. In addition, the sum of dimensions for a product of two compact metric spaces satisfies dim(X × Y) ≤ dim X + dim Y, with equality in many common cases. The concept is especially useful in dimension theory, manifold theory, and various applications in analysis and topology.