In this paper we apply the methods of supercompactifications and normal subbases to characterize subspaces of compact treelike spaces. This characterization is related to the subbase characterizations of ordered spaces, of trees and of normally supercompact spaces described in [5, 1, 9, 10, 14].

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.Michael [M] showed that for a continuum X the following statements are equivalent: (a) X has a selection, (b) X has a weak selection, and (c) X is orderable.In [Y], Young claims, without giving a proof, that statements (a), (b), and (c) are also equivalent for compact zero-dimensional spaces X.In this paper we will show that, for compacta, statements (a), (b), and (c) are always equivalent.1.The construction.Let X be compact and let s: X2 -» X be a weak selection.For each x E X define and Bx = {yEX\s(y,x)= y), Ax = { y E X | s(y, x) = x).Observe that both Ax and Bx are closed, that Ax u Bx = X and that Ax n Bx = 1.1.Theorem.Let X be a compact space.Then the following statements are equivalent:(a) X is orderable, (b) X has a weak selection, (c) X has a selection.