Inductive dimension

2020 Mathematics Subject Classification: Primary: 54F45 [MSN][ZBL]

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 [1]. The small inductive dimension was defined independently by P.S. Urysohn [2] and K. Menger [3]. 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.

References

[1]	L.E.J. Brouwer, "Ueber den natürlichen Dimensionsbegriff" J. Reine Angew. Math. , 142 (1913) pp. 146–152
[2]	P.S. Urysohn, "Les multiplicités cantoriennes" C.R. Acad. Sci. , 175 (1922) pp. 440–442
[3]	K. Menger, "Ueber die Dimensionalität von Punktmengen. I" Monatshefte Math. und Phys. , 33 (1923) pp. 148–160
[4]	P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian)

Comments

An extensive treatment of the subject can be found in [a1]. For a quick introduction to the dimension theory of separable metric spaces, see [a2], Chapt. 4.

References

[a1]	R. Engelking, "Dimension theory" , North-Holland & PWN (1978)
[a2]	J. van Mill, "Infinite-dimensional topology, prerequisites and introduction" , North-Holland (1988)