Die Invarianz des Zerlegens für kompakte Teilmengen von Produkten lineargeordneter Kontinuen (Q788317)

A linearly ordered continuum is a nonvoid linearly ordered set which has no first and no last element and which is dense and without gaps. So for each cut (A,B) A has a last or (aut) B has a first element. For a natural number n an n-continuum is a Cartesian product of n linearly ordered continua. In these we have the usual order topology and in their product the product topology. The latter generalizes \({\mathbb{R}}^ n\) in an order-theoretical manner. Several generalizations of classical invariance theorems of topology are proved: Let K and L be homeomorphic subsets of an n-continuum C. If K separates C then also L does so. (By the way, in the meantime I proved the more general theorem that for arbitrary homeomorphic compact subsets K, L of C the complements \(C\backslash K\) and \(C\backslash L\) have the same number of connectivity components.) This generalizes the separation theorem of Jordan-Brouwer-Alexander. Analogously to the Euclidean situation from the previous theorem the invariance of the open set and of dimension in n-continua can be proved. For the special case \(n=2\), \textit{U. Löttgen} had generalized the Jordan curve theorem to 2-continua [Über eine Verallgemeinerung des Jordanschen Kurvensatzes auf zweifach geordnete Mengen. Diss. (Köln 1952)], and also for \(n=2\) van Dalen had obtained the invariance of the open set for 2-continua [Finite products of locally compact ordered spaces. Diss. (Amsterdam 1972)]. Theorem 2 of the paper generalizes also the main result of the author's paper in Math. Ann. 203, 175-182 (1973; Zbl 0243.55011).