Several classes of hereditarily normal spaces are characterized in terms of extending upper semi-continuous compact-valued mappings. The case of controlled extensions is considered as well. Applications are obtained for real-valued semi-continuous functions.

The paper is devoted to generalizations of the Cencelj-Dranishnikov theorems relating extension properties of nilpotent CW complexes to their homology groups. Here are the main results of the paper: Theorem 0.1. Let L be a nilpotent CW complex and F the homotopy fiber of the inclusion i of L into its infinite symmetric product SP(L). If X is a metrizable space such that $X\tau K(H_k\left(L\right),k)$ for all k ≥ 1, then $X\tau K({\pi }_k\left(F\right),k)$ and $X\tau K({\pi }_k\left(L\right),k)$ for all k ≥ $.$Theorem 0.2. Let X be a metrizable space such that dim(X) < ∞ or X ∈ ANR. Suppose...

We deal with a hyperspace selection problem in the setting of connected spaces. We present two solutions of this problem illustrating the difference between selections for the nonempty closed sets, and those for the at most two-point sets. In the first case, we obtain a characterisation of compact orderable spaces. In the latter case --- that of selections for at most two-point sets, the same selection property is equivalent to the existence of a ternary relation on the space, known as a cyclic...