nLab
inductive dimension

Contents

Contents

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)ind(X) of a regular topological space XX is defined as follows: ind()=1ind(\emptyset)=-1 and ind(X)nind(X)\le n (n0n\ge0) if for any xXx\in X and any neighborhood VXV\subset X of xx there is an open subset UXU\subset X such that xUx\in U, UVU\subset V, and ind(U¯U)n1ind(\bar U\setminus U)\le n-1.

Large inductive dimension

The large inductive dimension (alias Brouwer–Čech dimension) Ind(X)Ind(X) of a normal topological space XX is defined as follows: Ind()=1Ind(\emptyset)=-1 and Ind(X)nInd(X)\le n (n0n\ge0) if for any closed subset AXA\subset X and any open subset VXV\subset X such that AVA\subset V there is an open subset UXU\subset X such that AUVA\subset U\subset V and Ind(U¯U)n1Ind(\bar U\setminus U)\le n-1.

Properties

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

notion of dimension

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)

Last revised on March 21, 2021 at 11:48:58. See the history of this page for a list of all contributions to it.