The hyperspace of closed subsets on the edge of irreducible continua

For a nonempty nowhere dense closed subset of a continuum $X$, consider the following properties: being a non-weak cut subset, a non-block subset, a weak non-block subset, a shore subset, a non-strong center, and a non-cut subset of X. In this paper, we provide necessary conditions for subsets of an irreducible continuum about a subset to have one of these properties, and we prove that these properties are equivalent for nonempty nowhere dense closed subsets of an irreducible continuum about a finite subset. This result completes the study previously conducted on non-cut points and for subsets on the edge of a continuum by several authors.