Boundary local connectivity of tiles in \(\mathbb R^2\) (Q861942)

Let \(C_p(r)\) be the circle with radius \(r>0\) centered at the point \(p\) in the plane, and \(A(p,r,R)\) the open annulus bounded by the two circles \(C_p(r)\) and \(C_p(R)\). In this paper the author shows that a continuum \(M\) in the plane is locally connected if and only if for every open annulus \(A(p,r,R)\) with \(p\in M\), the set \(A(p,r,R)\setminus M\) has at most finitely many components whose closures intersect the two circles \(C_p(r)\) and \(C_p(R)\). With this result, the author shows that the boundary of a connected and locally connected tile in the plane is a Peano continuum.