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On addition theorems for inductive dimensions
by
Vitali A. Chatyrko
Linköping University
Coauthors: M.G. Charalambous (University of the Aegean)
The problem discussed here is : Given a space X which is represented as the union of two subsets X1 and X2 of known dimension, what can be said about the dimension of X? Results giving an estimate of the dimension of the union of two subspaces are known as addition theorems.
There are classical addition theorems for dimensions ind and Ind if X is hereditarily normal. Namely, ind X <= ind X1 + indX2 and Ind X <= Ind X1 + IndX2. The inequalities are known as Menger-Urysohn formulas. Here we present different addition theorems for these dimensions in more general cases if Ind X1 = m and Ind X2 = n. For example, if X is normal then ind X <= 2(m+n+1).
The above result raises the problem of estimating ind X in terms of ind X1 and ind X2. In particular one question is whether ind X is finite when both ind X1 and ind X2 are finite. The answer is negative if instead of ind one considers inductive dimensions ind0 or Ind0 introduced by Charalambous and Filippov. In particular, a hereditarily normal compact space which is the union of two dense zero-dimensional subspaces can be infinite-dimensional in the sense of these dimensions.
Date received: June 19, 2002
Copyright © 2002 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 # caje-14.