On generalized rigidity (Q2725454)

The idea of the generalization of rigidity (and related properties) consists in replacement of the homeomorphisms (in a suitable definition) by a larger class of mappings. Now, let \(\mathcal M\) be a class of mappings between topological spaces that contains the class of homeomorphisms. A nondegenerate topological space is said to be: NEWLINENEWLINENEWLINE-- rigid with respect to \(\mathcal M\) if the only mapping in \(\mathcal M\) which maps \(X\) onto itself is the identity; NEWLINENEWLINENEWLINE-- chaotic with respect to \(\mathcal M\) provided that for any two distinct points \(p\) and \(q\) from \(X\) there exist an open neighbourhood \(U\) of \(p\) and an open neighbourhood \(V\) of \(q\) such that no open subset of \(U\) can be mapped onto any open subset of \(V\) by a mapping belonging to \(\mathcal M\). NEWLINENEWLINENEWLINEAmong others, it is proved that: (i) there exists a dendrite which is both rigid and rigid with respect to open mappings, (ii) there exists a dendrite which is both chaotic and not chaotic with respect to open mappings, (iii) there is no dendrite which is rigid with respect to monotone mappings.