arXiv:0906.4270v4 [math.GN] 7 Aug 2009 SELECTIONS, k-METRIZABLE COMPACTA, AND SUPEREXTENSIONS VESKO VALOV Abstract. A selection theorem for set-valued maps into spaces with binary normal closed subbases is established. This theo- rem implies some results of A. Ivanov (see [5], [6]) concerning superextensions of k-metrizable compacta. A characterization of k-metrizable compacta in terms of usco retractions into superex- tensions and extension of functions is also provided. 1. Introduction In this paper we assume that all topological spaces are Tychonoff and all single-valued maps are continuous. The following selection theorem for spaces with a binary normal closed subbase is established. Theorem 1.1. Let X be a space possessing a binary normal closed subbase S and Z an arbitrary space. Then every S -continuous set- valued map Φ: Z → X has a single-valued selection provided all Φ(z), z ∈ Z , are S -convex closed subsets of X . More generally, if A ⊂ Z is closed and every map from A to X can be extended to a map from Z to X , then every selection for Φ|A is extendable to a selection for Φ. Here, a collection S of closed subsets of X is called normal if for every S 0 ,S 1 ∈S with S 0 ∩ S 1 = ∅ there exists T 0 ,T 1 ∈S such that S 0 ∩ T 1 = ∅ = T 0 ∩ S 1 and T 0 ∪ T 1 = X . S is binary provided any linked subfamily of S has a non-empty intersection (we say that a system of subsets of X is linked provided any two elements of this system intersect). If S is a closed subbase for X and B ⊂ X , let I S (B)=  {S ∈S : B ⊂ S }. A subset B ⊂ X is called S -convex if for 1991 Mathematics Subject Classification. Primary 54C65; Secondary 54F65. Key words and phrases. continuous selections, Dugundji spaces, k-metrizable spaces, spaces with closed‘ binary normal subbases, superextensions. Research supported in part by NSERC Grant 261914-08. The paper was com- pleted during the author’s stay at the Faculty of Mathematics and Informatics (FMI), Sofia University, during the period June-July 2009. The author acknowl- edges FMI for the hospitality. 1