Fell continuous selections and topologically well‐orderable spaces

Abstract

Abstract. The present paper improves a result of [3] by showing that a space X is topologically well-orderable if and only if there exists a selection for F2(X) which is continuous with respect to the Fell topology on F2(X). In particular, this implies that F(X) has a Fell-continuous selection if and only if F2(X) has a Fell-continuous selection.

Hyperspace selections avoiding points

“…Here, X is an ordinal space if it is an ordinal equipped with the open interval topology. For some related results in the non-compact case, the interested reader is referred to [3,4,20,29,43]. The proof of Corollary 3.9 implies the following more restrictive property on the continuous selections for F (X).…”

“…To this end, let us recall that a point p ∈ X of a connected space X is cut if X \ {p} is not connected or, equivalently, if X \ {p} = U ∪ V for some subsets U, V ⊂ X with U ∩ V = {p}. Extending this interpretation to an arbitrary space X, a point p ∈ X was said to be cut [14], see also [6,13], if X \ {p} = U ∪ V and U ∩ V = {p} for some subsets U, V ⊂ X. Cut points were also introduced in [3], where they were called tie-points.…”

Ordinals and set-valued zero-selections for hyperspaces

“…Also, relying on the technique developed in [8], we extend this result to arbitrary scattered spaces in the following way:…”

Selections and countable compactness

“…Note that such sets U and V must be open because U = X \ V and V = X \ U . A concept similar to this played an important role for first countability of spaces X with se [F (X)] = ∅, see [8,14].…”