Inductive Dimension - Relationship Between Dimensions

Relationship Between Dimensions

Let be the Lebesgue covering dimension. For any topological space X, we have

if and only if

Urysohn's theorem states that when X is a normal space with a countable base, then

.

Such spaces are exactly the separable and metrizable X (see Urysohn's metrization theorem).

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. The Menger-Nöbeling theorem (1932) states that if X is compact metric separable and of dimension n, then it embeds as a subspace of Euclidean space of dimension 2n + 1. (Georg Nöbeling was a student of Karl Menger. He introduced Nöbeling space, the subspace of R2n + 1 consisting of points with at least n + 1 co-ordinates being irrational numbers, which has universal properties for embedding spaces of dimension n.)

Assuming only X metrizable we have (Miroslav Katětov)

ind X ≤ Ind X = dim X;

or assuming X compact and Hausdorff (P. S. Aleksandrov)

dim X ≤ ind X ≤ Ind X.

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 if and only if for every closed sub-space of the space and each continuous mapping there exists a continuous extension .