#### 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 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
{*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.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.40.2.14

**References:**

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

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

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

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

- E. Michael,
*Local properties of topological spaces*,
Duke Mat. J. **21** (1954), 163-171.

CrossRef

- L. Rubin,
*Relative collaring*,
Proc. Amer. Math. Soc. **55** (1976), 181-184.

CrossRef

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