Completions of spaces of functions on compact spaces with respect to the Hausdorff uniformity

Abstract

For bicompact X, the topology of uniform convergence on C(X) coincides with the Vietoris topology on the function graphs. The Vietoris topology on C(X) in turn is generated by the Hausdorff uniformity which is complete only for finite X. The supplement CH(X) of the space C(X) with respect to the Hausdorff uniformity consists of lower semi-continuous multivalued mappings Φ : X → ℝ with compact fibers. The paper studies the spaces CH(X) in the spaces of multivalued mappings. It is proved that CH(X) is a Q-manifold provided that X is a Peano continuum. Several results on the metrizability on bicompacta having the form CH(X, I) are also obtained. Bibliography: 27 titles.