Glasnik Matematicki, Vol. 44, No.2 (2009), 457-478.

COMPACTIFICATIONS OF [0,∞) WITH UNIQUE HYPERSPACE Fn(X)

Alejandro Illanes and Jorge M. Martinez-Montejano

Universidad Nacional Autónoma de México, Instituto de Matemáticas, Circuito Exterior, Cd. Universitaria, México, 04510, D.F. Mexico
e-mail: illanes@matem.unam.mx
e-mail: jorge@matem.unam.mx

Abstract.   Given a metric continuum X, Fn(X) denotes the hyperspace of nonempty subsets of X with at most n elements. In this paper we show the following result. Suppose that X is a metric compactification of [0,∞), Y is a continuum and Fn(X) is homemorphic to Fn(Y). Then: (a) if n ≠ 3, then X is homeomorphic to Y, (b) if n = 3 and the remainder of X is an ANR, then X is homeomorphic to Y. The question if the result in (a) is valid for n = 3 remains open.

2000 Mathematics Subject Classification.   54B20,54F15.

Key words and phrases.   Compactification, continuum, hyperspace, ray, symmetric product, unique hyperspace.

Full text (PDF) (free access)

DOI: 10.3336/gm.44.2.12

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