In the mathematical field of topology, the inductive dimension of a topological space X is either of two values, the small inductive dimension ind(X) or the large inductive dimension Ind(X).
These are based on the observation that, in n-dimensional Euclidean space Rn, (n − 1)-dimensional spheres (that is, the boundaries of n-dimensional balls) have dimension n − 1.
Therefore it should be possible to define the dimension of a space inductively in terms of the dimensions of the boundaries of suitable open sets.
The small and large inductive dimensions are two of the three most usual ways of capturing the notion of "dimension" for a topological space, in a way that depends only on the topology (and not, say, on the properties of a metric space).
For "sufficiently nice" spaces, the three measures of dimension are equal.
and every open set U containing x, there is an open set V containing x, such that the closure of V is a subset of U, and the boundary of V has small inductive dimension less than or equal to n − 1.
The Nöbeling–Pontryagin theorem then states that such spaces with finite dimension are characterised up to homeomorphism as the subspaces of the Euclidean spaces, with their usual topology.
, then it embeds as a subspace of Euclidean space of dimension
(Georg Nöbeling was a student of Karl Menger.
co-ordinates being irrational numbers, which has universal properties for embedding spaces of dimension
Assuming only X metrizable we have (Miroslav Katětov) or assuming X compact and Hausdorff (P. S. Aleksandrov) Either inequality here may be strict; an example of Vladimir V. Filippov shows that the two inductive dimensions may differ.
A separable metric space X satisfies the inequality