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) |
Inductive dimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inductive_dimension&oldid=40006