Topology Proceedings TOPOLOGY PROCEEDINGS SIZE LEVELS OF HYPERSPACES

Abstract. Let X be a metric continuum and C(X) the hyperspace of subcontinua of X. A size map is a continuous function σ : , where σ is a size map and It is known that size levels are subcontinua of C(X), so we consider the space SL(X) of size levels as a subspace of C(C(X)). In this paper we study the space SL(X) and obtain an intrinsic characterization of size levels. As a consequence, we show that SL ([0, 1]) is not homeomorphic to the Hilbert space l 2 and we obtain topological characterizations of size levels of the hyperspaces of [0, 1] and the unit circle in the plane