Continuous maps that reduce inductive dimensions
by
Jerzy Krzempek
Silesian University of Technology

It is well-known that the theorem on dimension-lowering maps for Ind is not true even for continuous maps between compact spaces. We present a new series of counter-examples in this compact case.

We modify a recent construction by V.A. Chatyrko, and for every pair of natural numbers m > n ≥ 1, obtain a compact space X_{m, n} such that
(a) ind X_{m, n} = Ind X_{m, n} = m, and
(b) every component of X_{m, n} is homeomorphic to the n-dimensional cube Iⁿ.
It follows that dim X_{m, n} = n. If we consider the decomposition D of X_{m, n} into components, then the quotient space X_{m, n}/D is zero-dimensional, and for the quotient map f: X_{m, n} → X_{m, n}/D, we have m = Ind X_{m, n} > Ind X_{m, n}/D + Ind f = n.

Date received: April 30, 2008