Inductive Dimension - Formal Definition

Formal Definition

We want the dimension of a point to be 0, and a point has empty boundary, so we start with

Then inductively, ind(X) is the smallest n such that, for every and every open set U containing x, there is an open V containing x, where the closure of V is a subset of U, such that the boundary of V has small inductive dimension less than or equal to n − 1. (In the case above, where X is Euclidean n-dimensional space, V will be chosen to be an n-dimensional ball centered at x.)

For the large inductive dimension, we restrict the choice of V still further; Ind(X) is the smallest n such that, for every closed subset F of every open subset U of X, there is an open V in between (that is, F is a subset of V and the closure of V is a subset of U), such that the boundary of V has large inductive dimension less than or equal to n − 1.