# Extensions of 2-point Selections

### From NZJM

**New Zealand Journal of Mathematics**

**Vol.** 38, (2008), **Pages** 1-8

**S. García-Ferreira**

Instituto de Matemáticas (UNAM)

Apartado Postal 61-3

Santa Maria

58089 Morelia

Michoacán

MEXICO

**V. Gutev**

School of Mathematical Sciences

Faculty of Science

University of KwaZulu-Natal

King George V Avenue

Durban 4041

SOUTH AFRICA

**T. Nogura**

Department of Mathematics

Faculty of Science

Ehime University

Matsuyama, 790-8577

JAPAN

mailto:nogura@dpc.ehime-u.ac.jp

**Abstract** We consider a special order-like relation on the subsets of a given space *X*, which is generated by a continuous selection *f* for at most 2-point subsets of *X*. The relation allows to define a "minimal" set of any non-empty compact subset of *X*, which is then used to construct continuous extensions of *f* over families of non-empty finite subsets of *X*. For instance, we show that *f* can be extended to a continuous selection for at most 3-point subsets if and only if the hyperspace of at most 3-point subsets has a continuous selection. Other possible applications are demonstrated as well.

**Keywords** Hyperspace topology, Vietoris topology, continuous selection, selection-regular selection

**Classification** (MSC2000) 54B20, 54C65