Extensions of 2-point Selections

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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

mailto:sgarcia@matmor.unam.mx

V. Gutev

School of Mathematical Sciences

Faculty of Science

University of KwaZulu-Natal

King George V Avenue

Durban 4041

SOUTH AFRICA

mailto:gutev@ukzn.ac.za

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

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