Topology Proceedings Document # baam-81

 Topology Proceedings 29 No. 1 (2005), pp. 377-384

# Extensional dimension and completion of maps

## H. Murat Tuncali, E. D. Tymchatyn and Vesko Valov

We prove the following completion theorem for closed maps between metrizable spaces: Let f: X ® Y be a closed surjection between metrizable spaces with e-dim f £ K, e-dim X £ LX, and e-dim Y £ LY for some countable CW-complexes K, LX, and LY. Then there exist completions [X\tilde] and [Y\tilde] of X and Y, respectively, and a closed surjection [f\tilde] : [X\tilde] ® [Y\tilde] extending f such that e-dim[f\tilde] £ K, e-dim[X\tilde] £ LX, and e-dim[Y\tilde] £ LY. We also establish a parametric version of a result of Miroslav Katetov characterizing the covering dimension of metrizable spaces in terms of uniformly 0-dimensional maps into finite-dimensional cubes.

Keywords: finite-dimensional spaces, regularly branched maps

Mathematics Subject Classification: 54F45 (55M10 54C65)

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