Selections and orderability

Abstract

Let X be a compact Hausdorff space. Then X has a selection if and only if X is orderable. 0. Introduction. Let I be a compact Hausdorff space and let 2X denote the hyperspace of nonempty closed subsets of X. A selection for A1 is a continuous map F: 2X -+ X such that F(A) E A for all A E 2X. Let X(2) denote the 2-fold symmetric product of X, i.e. the subspace of 2X consisting of all nonempty closed subspaces of X containing at most two points. A weak selection for X is a continuous map s: X(2) - X such that s(A) E A for all A E X(2). It is easy to see that X has a weak selection if and only if there is a continuous map s: X2 - X such that for all x, y EX, (1) s(x, y) = s(y, x), and (2)s(x,y) E {x,y}. Such a map s: X2 - X will also be called a weak selection.