### ${\mathcal{C}}_{p}-E$-movable and $\mathcal{C}-E$-calm compacta and their images

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We exhibit a metric continuum X and a polyhedron P such that the Cartesian product X × P fails to be the product of X and P in the shape category of topological spaces.

In the S-category $P$ (with compact-open strong shape mappings, cf. §1, instead of continuous mappings, and arbitrary finite-dimensional separable metrizable spaces instead of finite polyhedra) there exists according to [1], [2] an S-duality. The S-dual $DX,X=(X,n)\in P$, turns out to be of the same weak homotopy type as an appropriately defined functional dual $\overline{{\left({S}^{0}\right)}^{X}}$ (Corollary 4.9). Sometimes the functional object $\overline{{X}^{Y}}$ is of the same weak homotopy type as the “real” function space ${X}^{Y}$ (§5).

In this note we give an approach to shape covering maps which is comparable to that of *-fibrations (Mardesic and Rushing (1978)). The introduced notion conserves some important properties of usual covering maps.

Borsuk's quasi-equivalence relation on the class of all compacta is considered. The open problem concerning transitivity of this relation is solved in the negative. Namely, three continua X, Y and Z lying in ℝ³ are constructed such that X is quasi-equivalent to Y and Y is quasi-equivalent to Z, while X is not quasi-equivalent to Z.