Connectivity degrees of complements of closed sets in continua

Abstract: 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.