Implication between degrees n-Q2 and n-Qo

Determine whether, for any non-empty metric space X and any n in the natural numbers, the property n-Q2 implies the property n-Qo. Concretely, establish whether the existence of a finite subset B of X such that for each x in X there exists a continuum D contained in X with x in D and B intersect D nonempty necessarily implies the existence of a (possibly different) finite subset B' of X such that for every non-empty open set U in X there exists a continuum D contained in X with B' intersect int(D) nonempty and int(D) intersect U nonempty.

Background

The paper introduces a hierarchy of connectivity degrees for a space X, denoted n-Q1 through n-Q7 and n-Qo, that generalize several classical notions (e.g., non-cut, shore, and non-block points/sets). Here, n-Q2 and n-Qo are two such degrees defined via the existence of finite sets B that witness particular continuum-intersection properties: n-Q2 requires, for each point x, a continuum D containing x that meets B; n-Qo requires, for each non-empty open set U, a continuum D whose interior meets both U and B.

The authors map out known implications among these degrees and explicitly note a gap in understanding the relationship between n-Q2 and n-Qo. Resolving whether n-Q2 implies n-Qo would clarify the structural hierarchy and the relationships among the corresponding hyperspaces of non-cut sets (e.g., NWCn and NBOn).