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 less than or equal to K if e-dim f(-1)(y) less than or equal to K for every y is an element of Y. We prove that if e-dim f less than or equal to K then there exists a a-compact subset A of X such that e-dim A less than or equal to K and S\(X\A) is 0-dimensional. This result is an analogue for extensional dimension of a well-known theorem of Torunczyk. (C) 2000 Elsevier Science B.V. All rights reserved.