Dimension phenomena associated with βN-spaces

General theorems concerning s-connectedness and hyperspaces are first obtained. These results are applied to prove that: for a continuum X having zero surjective semispan, (1) each Whitney block in the hyperspace of subcontinua of X, C(X), has the fixed point property and (2) if f :Y→X is map from a continuum Y onto X, then the induced map f̂ :C(Y)→C(X) is universal. Both results provide new proofs to some theorems for arc-like continua. The first one answers a question asked by Nadler.