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) |
Inductive dimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inductive_dimension&oldid=13316