Glasnik Matematicki, Vol. 40, No.2 (2005), 339-345.

A LOCAL TO GLOBAL SELECTION THEOREM FOR SIMPLEX-VALUED FUNCTIONS

Ivan Ivanšić and Leonard R. Rubin

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 {U_α | α Γ} of X, so that for each α Γ, f_α : U_α → |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 f_α 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.

DOI: 10.3336/gm.40.2.14

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