Large inductive dimension

Definition
The large inductive dimension of a topological space is defined inductively. The empty set is assigned dimension $$-1$$. Suppose we have defined what it means for a topological space to have dimension $$\le m$$. Then a topological space $$X$$ has dimension $$\le m + 1$$ if given any two closed subsets $$A, B \subset X$$, there exists a closed subset $$P \subset X$$ of dimension $$\le m$$ such that the complement of $$P$$ is a disjoint union of open sets $$C$$ and $$D$$ where $$C$$ contains $$A$$ and $$D$$ contains $$B$$.

The large inductive dimension of $$X$$ is denoted $$Ind \ X$$.

Related notions

 * Small inductive dimension
 * Topological dimension