Transformations preserving tame sets

If X is a complex with a triangulation and if P is a homeomorph of a polyhedron in X with respect to this triangulation, then P is tame in X if there is a homeomorphism h of X onto itself and another triangulation of X in which h(P) is a polyhedron. A function from one complex X into a complex is called tame and is said to preserve tame sets if for each tame set PcX, f(P) is tame.

Tame local homeomorphisms from triangulated n-manifolds into triangulated n-manifolds and tame light open maps of 2-manifolds into themselves are homeomorphisms. Connected complexes are compact if and only if every tame map of the complex into itself has a polyhedral image. Tame linear maps of Euclidean spaces and tame simplicial maps on triangulated n-manifolds with boundaries are homeomorphisms if their images are of dimension greater than one.

Functions from polyhedra into topological spaces which take tame arcs onto sets consisting of finite number of components have images of, at most, a finite number of components. If the function and its inverse takes tame sets onto tame sets then the image is connected, provided its image is in a complex. If the function is from a topological space into a polyhedron, then it is continuous if and only if its inverse takes tame arcs onto closed sets. Finally a function from a complex to a complex is continuous if its inverse takes tame sets onto tame sets.

A function from an n-manifold into an n-manifold which has an image of dimension greater than one and which takes arcs onto arcs or points is a homeomorphism. A function from a compact triangulated n-manifold into a topological space which takes tame arcs onto arcs or points and whose image is not an arc or point is a homeomorphism. A function from a triangulated n-manifold into an n-manifold which takes tame arcs onto arcs or points and whose image is of dimension greater than one is a homeomorphism. A function from a triangulated n-manifold into a triangulated n-manifold which takes tame arcs onto connected tame sets such that the image of no tame arc contains a triod is a homeomorphism if its image set is not a point, arc or simple closed curve.

Finally there are tame maps which raise the dimension of sets. And there are 1:1 maps which do not preserve tame sets. A K-R manifold is a n-manifold with boundary whose interior is Eⁿ and whose boundary is Eⁿ⁻¹. A 1:1 map of a 2-dimensional K-R manifold onto a 2-dimensional K-R manifold is a homeomorphism.