Non-weak cut, non-block, shore, and non-strong center points of FₙK(X)

Given a continuum X and a positive integer n, Fₙ(X) denotes the hyperspace of all nonempty subsets of X with at most n points endowed with the Hausdorff metric. For K∈ Fₙ(X), Fₙ(K,X) denotes the set of all elements of Fₙ(X) containing K. We will consider FₙK(X) the quotient space obtained from Fₙ(X) by shrinking Fₙ(K,X) to one point set, endowed with the quotient topology. In this paper, we study the relationship between some types of non-cut points of FₙK(X) and the condition of being of the same type of non-cut set over its preimages in Fₙ(X) under the natural quotient map. The non-cut type sets considered here are: non-weak cut, non-block, shore, and non-strong center sets.