On snakelike bicompacta (Q1060456)

A space X is called a snake-like continuum if for any open covering there exists a finite open refinement \(\nu =\{{\mathcal U}_ 1,{\mathcal U}_ 2,...,{\mathcal U}_ s\}\) for which \({\mathcal U}_ i\cap {\mathcal U}_ j\neq \emptyset\) iff \(| i-j| \leq 1\). It is clear that dim X\(=1\). Theorem 1. There exists a snake-like continuum \({\mathcal X}\) which satisfies the first axiom of countability for which ind \({\mathcal X}=Ind {\mathcal X}=2\) and moreover \(ind_ x{\mathcal X}=2\) for each \(x\in {\mathcal X}\). In the second part the author proves the following Theorem. For any ordinal number \(\alpha \leq \omega_ 1\) there exists a compact space \(Y_{\alpha}\) for which \(\sigma\) ind \(Y_{\alpha}=\alpha\).