Concerning aposyndetic and non-aposyndetic continua

Certain Homogeneous Unicoherent Indecomposable Continua

“…Furthermore, G is upper-semicontinuous for if some sequence Xi, x2, x3, • • ■ of points of distinct elements of G converged to a point x of an element of G, say H, but some infinite sequence yit y2, y3, ■ ■ • of points from the same elements of G converged to a point y of M-H, then M would be aposyndetic at y with respect to x. But for each i, M is not aposyndetic at y,-with respect to x<, and this contradicts Theorem 1 of [7]. So G is upper-semicontinuous.…”

“…Proof.3 Suppose that U is an open subset of M and H is a subset of M -U such that in order for a point x to belong to H it is necessary and sufficient that Ux= U. In case M contains no such sets U and H, M is indecomposable by Theorem 9 of [7].…”

“…So w belongs to H. 3 Throughout this proof M will be considered to be space. If there do not exist points x and z of M such that M is aposyndetic at z with respect to x, then M is indecomposable (Theorem 9 of [7]). So because M is homogeneous it will be assumed that for each point x of M, Ux exists (that is, nonvacuous).…”

Non-aposyndesis and non-hereditary decomposability

“…Essentially indecomposable sets* Theorem 9 of [1] states that a necessary and sufficient condition that the continuum M be indecomposable is that if x and y are points of M then M is nonaposyndetic at x with respect to y. We obtain, from this condition, the following definition.…”

“…If H is a subcontinuum of M containing p in its interior, then for some i^> N the point x t is in Int (H) and therefore Int (K(Xi)) c ff. Let D be the closed square disk in E 2 whose opposite vertices are (-1, -1) and (1,1). Let D\ and D\ be subsets of D homeomorphic to D -{(1, 0)} which spiral out to Bd (JD), as indicated in Figure 1, so that Bd (D) is the limiting set of each of the spirals.…”