Selections and deleted symmetric products

Abstract

We give a very simple example of a connected second countable space X whose hyperspace ½X nþ1 of unordered ðn þ 1Þtuples of points has a continuous selection, but ½X n has none.This settles an open question posed by Michael Hrus ˇa ´k and Ivan Martı ´nez-Ruiz. The substantial part of the paper sheds some light on this phenomenon by showing that in the presence of connectedness this is essentially the only possible example of such spaces.

Hyperspace Selections Avoiding Points

“…If p ∈ X is a cut point of X, then there are disjoint sets U, V ⊂ X such that X \ {p} = U ∪ V and {p} = U ∩ V . Following [3], such a pair (U, V ) of sets will be called a p-cut of X. A point p ∈ X is said to separate x, y ∈ X if x ∈ U and y ∈ V for some p-cut (U, V ) of X.…”

“…In these terms, X is called almost weakly orderable [3, Definition 3.2] if it has finitely many noncut points and among every three points of X with two of them being cut, there is one which separates the other two. If X is almost weakly orderable, then there exists a partial order ≤ on X such that two points of X are ≤-comparable precisely when they can be separated, moreover this order is compatible with the topology of X in the sense that all ≤-open intervals are open in X, see [3,Corollary 3.7]. Such a partial order on X is called a separation partial order, and any two separation partial orderings on X are either identical or inverse to each other [3, Proposition 3.8].…”