Local connectivity from equality of hyperspaces CC1(X) or NWC1(X) with NC1(X)

Ascertain whether a continuum X must be locally connected if CC1(X) equals NC1(X), or if NWC1(X) equals NC1(X). In particular, determine whether there exists a non-locally connected continuum X with NC1(X) = NWC1(X), or with CC1(X) = NC1(X).

Background

The paper connects properties of the original continuum X with the structure of hyperspaces of non-n-cut sets. For locally connected continua, the authors prove CC1(X) = NC1(X), and more generally CCn(X) = NCn(X) for all n.

They note the absence of a known non-locally connected example with NC1(X) = NWC1(X) and raise whether such hyperspace equalities compel local connectedness of X. A partial answer is provided for irreducible continua (Corollary 5.18), where NC1(X) = NWC1(X) holds if and only if X is an arc.