On sum and product theorems for dimension Dind.
by
V. A. Chatyrko
Linköping
Coauthors: B. A. Pasynkov (Moscow)

All our spaces will be normal Hausdorff. The dimension function Dind has been introduced by A. Arhangel'skij in the following way: 1) Dind X = -1 iff X = \emptyset;
2) Dind X <= n >= 0 if for any finite open cover \nu of X we can find a disjoint family \mu of open subsets of X such that Dind ( X \ \cup \mu) <= n-1 and \mu\succ \nu. ( cf. [1]).

It was known [1] that IndX <= Dind X. We prove that Dind is finite if Ind is finite. More precisely, for every space X with Ind X >= 1 we have

Dind X <= (Ind X+2)! / 6.

We establish various sum theorems for Dind. For example,

Let a space X be the union of finite many non-empty closed subspaces X₁, ..., X_n, n >= 3, such that the sets X_i \X₁, i = 2, ..., n, are disjoint. Then

Dind X <= Dind X1 + max { Dind Xi : i = 2, ..., n}.

The last inequality leads to essentially better estimations for dimension Dind of topological products than for Ind from [2]. One of them is

For any non-empty locally compact paracompactum Y and a space X such that the product X x Y is normal and Ind X + Ind Y >= 2 we have

Dind (X x Y) <= 3/2 2( Dind X + Dind Y ) - 2.

References

[1] V.Egorov and Ju.Pristavkin, On a definition of dimension, Soviet Math. Dokl. 9 (1968), 188-191.

[2] B.A. Pasynkov, On the finite-dimensionality of topological products, Topol. Appl. 82 (1998), n. 1-3, 377-386.

Date received: June 12, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeu-18.