An AR-map Whose Range is More Infinite-Dimensional Than Its Domain

We construct an example of an AR-map f : X → Y f:X \to Y , where X X is a strongly countable dimensional compact AR and Y Y is a countable dimensional AR which is not strongly countable dimensional. Using this map we find a shrinkable decomposition of the pre-Hilbert space l f 2 l_f^2 whose quotient map does not stabilize to a near homeomorphism. We also present a partial result concerning the question whether cell-like maps preserve countable dimensionality.