Inductive dimension

Definition

Let $X$ be a topological space, and write $\partial V$ for the boundary of a set $V \subseteq X$. The small inductive dimension $\operatorname{ind} X$ is defined by induction on $n \ge -1$:

1. $\operatorname{ind} X = -1$ if and only if $X = \emptyset$.
2. For $n \ge 0$, $\operatorname{ind} X \le n$ if for every point $x \in X$ and every open set $U$ with $x \in U$ there is an open set $V$ with $x \in V \subseteq U$ and $\operatorname{ind} \partial V \le n - 1$.
3. $\operatorname{ind} X = n$ if $\operatorname{ind} X \le n$ and not $\operatorname{ind} X \le n - 1$; and $\operatorname{ind} X = \infty$ if $\operatorname{ind} X \le n$ holds for no $n$.

The large inductive dimension $\operatorname{Ind} X$ is defined in the same way, with a new clause 2. For $n \ge 0$, $\operatorname{Ind} X \le n$ if for every closed set $A$ and every open set $U \supseteq A$ there is an open set $V$ with $A \subseteq V \subseteq U$ and $\operatorname{Ind} \partial V \le n - 1$. Here the boundary $\partial V$ carries the subspace topology.