The small and large inductive dimension are two dimension terms considered in the mathematical sub-area of topology . These terms do not use any algebraic constructions to define a dimension, as is known from the theory of vector spaces , but only the topological space considered itself. It is an alternative to Lebesgue's coverage dimension , which is denoted by and used here for comparison purposes becomes.
The idea of the inductive dimension is based on the observation that the edge of a -dimensional sphere is -dimensional, whereby -dimensional is to be understood here in the sense of differential geometry (see manifold ) or simply purely graphically. This suggests the idea of tracing the term dimension of a set back to the term dimension of the edge of this set and thus striving for an inductive definition.
Since a one-point space, which should definitely have the dimension 0, has an empty border, the dimension of the empty set must be defined as −1. An implementation of the idea of inductive definition then leads to the following two variants:
definition
The small inductive dimension
The small inductive dimension of a topological space is defined as follows:
if for every point and every open environment of there is an open environment of with and .
That explains what . One further defines:
if and not
if the inequality holds for none .
The great inductive dimension
If one replaces the point from the definition of the small inductive dimension by any closed set , one obtains the concept of the large inductive dimension. More precisely: The large inductive dimension of a topological space is defined as follows:
if there is an open environment of with and for every closed set and every open environment of .
That explains what . One further defines:
if and not
if the inequality holds for none .
Remarks
Since the single-point subsets are closed in -rooms, it follows immediately for such spaces .
The statement can be reformulated as follows: Every point has a surrounding basis made up of closed sets with edges of the small inductive dimension . In particular, in this case each point has an environment base made up of closed sets, so that this term only makes sense in regular spaces .
The statement can be reformulated as follows: For every two disjoint , closed subsets there are open environments and with , and . In particular, in this case two disjoint, closed sets can be separated by open sets , so that this term only makes sense in normal spaces .
While with the small inductive dimension each point of the room can be assigned a dimension in an obvious way, this is not possible with the large inductive dimension, this relates to the entire room.
The large inductive dimension is sufficient for the sum theorem for completely normal rooms , that is
CH Dowker : Is completely normal and a sequence of closed sets with , then applies .
For general normal rooms, the sums rate applies neither for nor for , not even if one restricts oneself to compact Hausdorff rooms .
Product set
It is said that a dimension concept fulfills a product proposition if the dimension of the product space of two spaces can be estimated against the sum of the dimensions of these two spaces. Notice .
If and are not empty regular Hausdorff rooms, then applies .
If perfectly normal and metrizable and both are non-empty, then the following applies .
An analogous statement applies to the coverage dimension if and both are metrizable or if they are paracompact and compact.
literature
Keiô Nagami: Dimension Theory (= Pure and Applied Mathematics. Vol. 37). Academic Press, New York NY et al. 1970, ISBN 0-12-513650-1 .