nLab inductive dimension

Idea

The small and large inductive dimensions, together with the Lebesgue covering dimension are the three main notions of dimension for topological spaces. They coincide in some, but not all, cases.

Small inductive dimension

The small inductive dimension (alias Menger–Urysohn dimension) $ind(X)$ of a regular topological space $X$ is defined as follows: $ind(\emptyset)=-1$ and $ind(X)\le n$ ($n\ge0$) if for any $x\in X$ and any neighborhood $V\subset X$ of $x$ there is an open subset $U\subset X$ such that $x\in U$, $U\subset V$, and $ind(\bar U\setminus U)\le n-1$.

Large inductive dimension

The large inductive dimension (alias Brouwer–Čech dimension) $Ind(X)$ of a normal topological space $X$ is defined as follows: $Ind(\emptyset)=-1$ and $Ind(X)\le n$ ($n\ge0$) if for any closed subset $A\subset X$ and any open subset $V\subset X$ such that $A\subset V$ there is an open subset $U\subset X$ such that $A\subset U\subset V$ and $Ind(\bar U\setminus U)\le n-1$.

Properties

The subspace theorem: for every subspace $M$ of a regular space $X$ we have $ind(M)\le ind(X)$. (Theorem 1.1.2 in Engelking95.)

Related concepts

notion of dimension

• homotopy dimension

• cohomology dimension (virtual)

• covering dimension

• Heyting dimension

• dimension of a manifold

• dimension of a cell complex

• dimension of a separable metric space

• small inductive dimension of a regular space

• large inductive dimension of a normal space

References

• Ryszard Engelking, Dimension Theory, Mathematical Library 19, North-Holland Publishing/Polish Scientific Publishers 1978 (pdf)

• Ryszard Engelking, Theory of Dimensions – Finite and Infinite, Sigma Series in Pure Mathematics 10, Helderman 1995 (pdf)