A fundamental well-known result about selectors says that a strongly zero- dimensional metric space has a selector on closed sets iff it is a completely metrizable space [3,4]. One may wonder whether a similar result can be formulated in non-Archimedean spaces. A non-Archimedean space is a space which has a base B for the open sets which is a tree with respect to reverse inclusion.
Recently it has been proved that a non-Archimedean space with uncountable punctual character at any non-isolated point has a selector on closed sets iff it is scattered [2].
By defining a completeness condition related to the tree B, we shall construct an interesting class of closed sets which admits a selector.
Definition A subset Y of a non-Archimedean space is said to be B-complete if, for every chain of elements of B which meet Y, their intersection has a point in Y.
B-completeness in \omega\mu-metric spaces is studied in [1], where some problems about selections for l.s.c. multivalued functions is discussed.
Theorem Let X be a non-Archimedean space. Then there exists a selector on B-complete subsets of X.
Corollary There exists a selector on the finite subsets of any non-Archimedean space.
Let M denote the Michael line. M is a non-Archimedean space which has a base B of height \omega+1. Let D be the dense set consisting of the isolated points which are elements of finite height in the tree B.
Theorem The following conditions are equivalent for a subset E of M:
In the Euclidean topology, E is a closed set without limit points in D.
It is well-known [3] that there is no selector on the closed discrete subsets of R.
Corollary On the discrete closed sets of R there exists a selector which is continuous for the Vietoris topology induced by the Michael line topology.
References
Date Received: February 1, 2000