# Inductive dimension

large inductive dimension $\mathrm{Ind}\,X$, small inductive dimension $\mathrm{ind}\,X$
Dimension invariants of a topological space $X$; both are defined by means of the notion of a partition between two sets. The definition is by induction, as follows. For the empty space $X = \emptyset$ one sets $\mathrm{Ind}\,\emptyset = \mathrm{ind}\,\emptyset = -1$. Under the hypothesis that all spaces $X$ for which $\mathrm{Ind}\,X < n$ are known, where $n$ is a non-negative integer, one puts $\mathrm{Ind}\,X < n+1$ if for any two disjoint closed subsets $A$ and $B$ of $X$ there is a partition $C$ between them for which $\mathrm{Ind}\,C < n$. Here, a closed set $C$ is called a partition between $A$ and $B$ in $X$ if the open set $X \setminus C$ is the sum of two open disjoint sets $U_A$ and $U_B$ containing $A$ and $B$, respectively. This definition transfers to the definition of small inductive dimension $\mathrm{ind}\,X$ by taking one of the sets $A$ or $B$ to consist of a single point, while the other is an arbitrary closed set not containing this point.
The large inductive dimension was defined for a fairly wide class of (metric) spaces by L.E.J. Brouwer . The small inductive dimension was defined independently by P.S. Urysohn  and K. Menger . The study of inductive dimensions and, more generally, of dimension invariants, is only of interest under the hypothesis that the space $X$ satisfies sufficiently strong separation axioms, mainly the axiom of normality.