Finite decompositions and the depth of a continuum. (English) Zbl 0713.54038

Let X be a decomposable metric continuum. We call an n-decomposition of X every family \(\{X_ 1,X_ 2,...,X_ n\}\), \(n\geq 2\), of proper subcontinua of X such that \(X=\cup^{n}_{i=1}X_ i\). Fix an \(n\geq 2\); by \(I_ n(X)\) we mean the set of all subcontinua P of X such that, for each n-decomposition of X, there exists a member \(X_{i_ 0}\) of the decomposition such that \(P\subset X_{i_ 0}\). We put \(I(X)=\cap^{\infty}_{n=2}I_ n(X)\); see also [S. D. Iliadis, Vestnik Moskov. Univ., Ser. I 29, Nr. 6, 60-65 (1974; Zbl 0294.54030), Engl. transl. in Moscow Univ. Math. Bull. 29, Nr. 5/6, 94-99 (1974)]. In a hereditarily decomposable continuum X, a well ordered sequence \(\{X_{\alpha}\}\), respectively \(\{X^ n_{\alpha}\}\), of nondegenerate subcontinua of X is said to be normal, respectively n- normal, if and only if (i) \(X_ 1=X\), respectively \(X^ n_ 1=X\); (ii) \(X_{\alpha +1}\in I(X_{\alpha})\), respectively \(X^ n_{\alpha +1}\in I_ n(X^ n_{\alpha})\); and (iii) \(X_{\beta}=\cap_{\beta '<\beta}X_{\beta '}\), respectively \(X^ n_{\beta}=\cap_{\beta '<\beta}X^ n_{\beta '}\), for any limiting ordinal number \(\beta\). We define the depth, respectively n-depth, of a hereditarily decomposable continuum X to be a certain ordinal number which we denote by k(X), respectively by \(k_ n(X)\). Namely, this is the minimum ordinal number a for which the order-type of every normal, respectively n-normal, sequence is not greater than a.
We investigate the relation between the notions of aposyndesis, convergence continua, finite decompositions, and finally the notion of the depth. We give some partial answers to the question under what conditions the equality \(k(X)=k_ n(X)\) holds for each \(n\geq 2\). By definitions of \(I_ n(x)\), \(n\geq 2\), and I(X), we have, for each decomposable continuum X, the following inclusions: \[ I_ 2(X)\supseteq I_ 3(X)\supseteq...\supseteq I_ n(X)\supseteq...\supseteq I(X). \] We show that none of these inclusions can be replaced by equality.


54F15 Continua and generalizations
54F55 Unicoherence, multicoherence


Zbl 0294.54030