### Ivan Ivanšić and Leonard R. Rubin

Department of Mathematics, University of Zagreb, Unska 3, P.O. Box 148, 10001 Zagreb, Croatia
e-mail: ivan.ivansic@fer.hr

Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, USA
e-mail: lrubin@ou.edu

Abstract.   Suppose we are given a function σ : X K where X is a paracompact space and K is a simplicial complex, and an open cover {Uα | α Γ} of X, so that for each α Γ, fα : Uα |K| is a map that is a selection of σ on its domain. We shall prove that there is a map f : X |K| which is a selection of σ. We shall also show that under certain conditions on such a set of maps or on the complex K, there exists a σ : X K with the property that each fα is a selection of σ on its domain and that there is a selection f : X |K| of σ. The term selection, as used herein, will always refer to a map f, i.e., continuous function, having the property that f(x) σ(x) for each x in the domain.

2000 Mathematics Subject Classification.   54C65, 54C05, 54E20.

Key words and phrases.   Contiguous functions, continuous function, discrete collection, infinite simplex, K-modification, locally finite-dimensional complex, paracompact, polyhedron, principal simplex, selection, simplex, simplicial complex.

Full text (PDF) (free access)

DOI: 10.3336/gm.40.2.14

References:

1. R. Engelking, General Topology, PWN-Polish Scientific Publishers, Warsaw, 1977.

2. I. Ivansic and L. Rubin, A selection theorem for simplex-valued maps, Glas. Mat. Ser. III 39(59) (2004), 331-333.

3. S. Mardesic, Extension dimension of inverse limits, Glas. Mat. Ser. III 35(55) (2000), 339-354.

4. S. Mardesic, Extension dimension of inverse limits. Correction of a proof., Glas. Mat. Ser. III 39(59) (2004), 335-337.

5. E. Michael, Local properties of topological spaces, Duke Mat. J. 21 (1954), 163-171.
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6. L. Rubin, Relative collaring, Proc. Amer. Math. Soc. 55 (1976), 181-184.
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