Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces

Details Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces

2008-03-29

arxiv.org/abs/0803.4243v1

Topology and Its Applications 155:8 (2008), 764-772.

Mathematics

General Topology

Scientific paper

10.1016/j.topol.2007.02.014

Our main result states that the hyperspace of convex compact subsets of a compact convex subset $X$ in a locally convex space is an absolute retract if and only if $X$ is an absolute retract of weight $\le\omega_1$. It is also proved that the hyperspace of convex compact subsets of the Tychonov cube $I^{\omega_1}$ is homeomorphic to $I^{\omega_1}$. An analogous result is also proved for the cone over $I^{\omega_1}$. Our proofs are based on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved.

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