Topology Proceedings 30 No. 1 (2006), pp. 1-23

Homogeneous Circle-Like Continua Are C-Determined

Gerardo Acosta

For a metric continuum X we denote by C(X) the hyperspace of subcontinua of X with the Hausdorff metric. A class G of continua is said to be C-determined provided that if X, Y ∈ G and the hyperspaces C(X) and C(Y) are homeomorphic, then continua X and Y are homeomorphic. In 1978, Sam B. Nadler, Jr. in [Hyperspaces of Sets. Monographs and Textbooks in Pure and Applied Mathematics, Vol. 49] asked if the class of circle-like continua is C-determined. In this paper, we provide a partial positive answer to this question by showing that both the class of arcwise connected circle-like continua and the class of homogeneous circle-like continua are C-determined. By considering Knaster continua, we present two other classes of circle-like continua which are C-determined.

Keywords: arc-like continuum, C-determined, circle-like continuum, homogeneous space, Knaster continua, manifold interior, semi-boundary, unique hyperspace

Mathematics Subject Classification: 54{B}20 (54{B}15 54{F}15 54{F}50)

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Topology Proceedings, Volume 30, No. 1 (2006)
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