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Let ConvF(Rn) be the space of all non-empty closed convex sets in Euclidean space Rn endowed with the Fell topology. In this paper, we prove that ConvF(Rn) … Rn £ Q for every n > 1 whereas ConvF(R) …R £ I. Let Conv(X) be the set of all non-empty closed convex sets in a normed linear space X = (X;k¢k). We can consider various topologies on Conv(X). In the paper (6), the AR-property of the spaces Conv(X) with the Hausdor metric topology, the Attouch-Wets topology, and the Wijsman topology has been studied. In this paper, we shall consider the Fell topology on Conv(X), which is generated by the sets of the form U¡ = fA 2 Conv(X) j A \ U 6= ;g and (X n K)+ = fA 2 Conv(X) j A ‰ X n Kg; where U is open and K is compact in X. This topology is also defined on the set Conv⁄(X) = Conv(X) ( f;g. By Conv⁄F(X) and ConvF(X), we denote the spaces Conv⁄(X) and Conv(X) admitting the Fell topology. In case X is finite-dimensional (equivalently locally compact), ConvF(X) is a locally compact metrizable space and Conv⁄F(X) is its Alexandor one- point compactification. It is easy to see that ConvF((0;1)) is homeomorphic to (…) the triangle with two vertices removed, ¢ n f(0;0);(1;1)g, where ¢ = f(x;y) 2 I2 j x 6 yg ‰ I2. Since ConvF(R) … ConvF((0;1)), we have ConvF(R) … ¢ n f(0;0);(1;1)g …R £ I;

The Spaces of Closed Convex Sets in Euclidean Spaces with the Fell Topology