Connectivity degrees of complements of closed sets in continua

In the literature, various types of points and meager sets whose complements are connected have been studied, such as colocally connected points, non-weak cut points/sets, non-block points/sets, shore points/sets, etc. We extend that study, in the following way: considering a continuum $X$ and a natural number $n$, we investigate sets $A \in 2^X$ meeting the criterion that $X - A$ has at most $n$ components, and we introduce degrees of connectivity of the complement of $A$. When $n=1$ and $A$ is meager or singleton, these new definitions are equivalent to the known definitions of non-cut points/sets.