Abstract
Suppose that X X is a Hausdorff space such that its Vietoris hyperspace ( F ( X ) , τ V ) ({\mathcal {F}}(X),\tau _{V}) has a continuous selection. Do disconnectedness-like properties of X X depend on the variety of continuous selections for ( F ( X ) , τ V ) ({\mathcal {F}}(X),\tau _{V}) and vice versa? In general, the answer is “yes” and, in some particular situations, we were also able to set proper characterizations.
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