On the Possibility of Joining Points with Paths Inside Balls

The article is devoted to the possibility of joining by paths the points of two sequences that converge to some (one) boundary point of a domain of \(n\)-dimensional Euclidean space. It is well known that in domains locally connected on their boundary, it is possible to construct a system of infinitesimal neighborhoods of the boundary point, the intersection of which with a given domain is path connected. Of course, this construction is impossible in an arbitrary domain, for example, it is not true in a disk with a cut on the plane. The problem of the possibility of joining pairs of points in the intersection of a domain with arbitrarily small concentric balls seems more subtle. The problems that are solved in this manuscript are of an even more subtle nature. Let us pose the following problem: let some set be nowhere dense, and a given domain is finitely connected with respect to this set, and balls centered at a fixed point of the boundary form connected intersections with the given domain, and the intersections of these balls with the complement of the specified set consist of no more than \(m\) components. In this case, is it possible to join the points of sequences converging to some boundary point by paths that lie in the intersection of the domain with infinitesimal balls centered at this point, which contain no more than \(m-1\) points of the aforementioned set? The answer to this question is positive and is given in the article. It should be noted that relatively recently we have shown the existence of infinitesimal neighborhoods with the specified property for paths that join the points of neighborhoods and intersect with a “bad” set no more than \(m-1\) times. The purpose of our manuscript is to prove that under some additional conditions one can take not an abstract system of neighborhoods, but a sequence of concentric balls. To a certain extent, the proof of this result is constructive. In particular, the possibility of joining points by paths that contain at most \(m-1\) points from some nowhere dense set is related to finite connectedness with respect to the complement of this set. This construction has applications to the problem of Hölder continuity of maps in a given domain. In particular, the existence of families of paths with a conditionally “minimal” intersection with some nowhere dense set is related to the study of non-closed maps, or maps that do not preserve the boundary of the domain.