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 $\leq m$. Then a topological space $X$ has dimension $\leq m+1$ if given any two closed subsets $A,B\subset X$, there exists a closed subset $P\subset X$ of dimension $\leq m$ 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