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

### A SELECTION THEOREM FOR SIMPLEX-VALUED MAPS

### Ivan Ivanšić and Leonard R. Rubin

I. Ivansic, Department of Mathematics, University of Zagreb,
Unska 3, P.O. Box 148, 10001 Zagreb, Croatia

*e-mail:* `ivan.ivansic@fer.hr`

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.

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