Verallgemeinerung des Zerlegungssatzes von Jordan-Brouwer-Alexander auf Produkte lineargeordneter Kontinuen. (Generalization of the Jordan- Brouwer-Alexander decomposition theorem to products of linearly ordered continua) (Q1086878)

The author considers linearly ordered connected spaces with no first or last element and generalizes to finite products of such spaces, \(C=C_ 1\times C_ 2\times...\times C_ n\) certain positional properties of compact subspaces K of \(R^ n\). He has a notion of degree of a map and uses it to prove that if K is a compact subset of C and f:K\(\to C\) is continuous and injective, then C-K and C-f(K) have the same number of components. Also, if A is the closure of a bounded open set and \(f: A\to C\) is such that \(f(x)=x\) for \(x\in \partial A\), then \(A\subset f(A)\) (a no retraction theorem). In [Arch. Math. 42, 161-167 (1984; Zbl 0531.54040)] he proved an invariance of domain theorem.