Glasnik Matematicki, Vol. 39, No.2 (2004), 331-333.

A SELECTION THEOREM FOR SIMPLEX-VALUED MAPS

Ivan Ivanšić and Leonard R. Rubin

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

Abstract.   The purpose of this short note is to prove the following theorem. Let X be a hereditarily normal paracompact Hausdorff space, K be a simplicial complex, and σ : X → K be a function. Suppose that {U_α | α ∈ Γ} and {f_α | α ∈ Γ} are collections such that for each α ∈ Γ, f_α is a map of U_α to |K|, and if x ∈ U_α, then f_α(x) ∈ σ(x). Assume further that {U_α | α ∈ Γ} is an open cover of X. Then there exists a map f : X → |K| such that for each x ∈ X, f(x) ∈ σ(x).

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

Key words and phrases.   Contiguous functions, continuous function, hereditarily paracompact, polyhedron, selection, simplex, simplicial complex, stratifiable space.