Inductive dimension

large inductive dimension , small inductive dimension

Dimension invariants (cf. Dimension invariant) of a topological space ; 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 one sets . Under the hypothesis that all spaces for which are known, where is a non-negative integer, one puts if for any two disjoint closed subsets and of there is a partition between them for which . Here, a closed set is called a partition between and in if the open set is the sum of two open disjoint sets and containing and , respectively. This definition transfers to the definition of small inductive dimension by taking one of the sets or 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 satisfies sufficiently strong separation axioms, mainly the axiom of normality (cf. Separation axiom).

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)