On extensional dimension of maps

Let K be a CW-complex. A map f :X→Y of compacta X and Y is said to be of e-dim ≤K if e-dim f−1(y)≤K for every y∈Y. We prove that if e-dim f≤K then there exists a σ-compact subset A of X such that e-dim A≤K and f|X\A is 0-dimensional. This result is an analogue for extensional dimension of a well-known theorem of Torunczyk.