On continuum-wise minimality
We say that a homeomorphism of a compact metric space is cw-minimal if all the proper closed invariant subsets have dimension zero. This concept was previously considered by H. Kato. We explore this notion and provide examples. We give sufficient conditions for the existence of cw-minimal subsets and we prove several characterizations. We show that cw-minimal systems are transitive and either minimal or sensitive if the space is locally connected. A subset is said to be mindual if it intersects every minimal subset. We show that every cw-minimal subset contains a closed, zero-dimensional mindual set.