Universal space in the Cartesian product of Peano curves without free arcs
A nontrivial continuum \(K\) is a Peano curve (without free arcs) if it is a \(1\)-dimensional locally connected metric compact space (such that no open subset of \(K\) is homeomorphic to an interval). Let \(K_1, \dots, K_{n+1}\), \(n \geq 1,\) be Peano curves without free arcs. The author shows that in the Cartesian product \(K_1 \times \dots \times K_{n+1}\) there exists a universal space \(T_n\) for the class of \(n\)-dimensional separable metric spaces. Moreover, for every separable metric space \(X\) satisfying dim \(X \leq n\) the set of embeddings of \(X\) into \(T_n\) is a residual set in \(C(X, K_1 \times \dots \times K_{n+1}),\) and the set of embeddings of \(X\) into \(K_1 \times \dots \times K_n \times I\), where \(I\) is a closed bounded interval, is a dense set in \(C(X, K_1 \times \dots \times K_n \times I)\). Some additional results concern extensions of mappings. One of them is the following. For a given separable metric space \(X\), a given Peano curve \(K\) without free arcs and a countable family of mappings \(\{f_i : X \to X\}_{i=1}^\infty\) the set of embeddings \(h: X \to K^\omega\) such that there exist extensions \(\{f^*_i : \overline{h(X)} \to \overline{h(X)}\}_{i=1}^\infty\) of the maps \(\{hf_ih^{-1}\}_{i=1}^\infty\) is a residual set in \(C(X, K^\omega).\)
scientific article; zbMATH DE number 5706279
scientific article