  Hilbert initiated the standpoint in foundations of mathematics. From this
standpoint, we allow only a finite number of repetitions of elementary
operations when we construct objects and morphisms. When we start from a subset
of a Euclidean space. Then we assume that any element of the line has only a
finite number of connected components. We call the set tame if the assumption
is satisfied, and define a tame morphism in the same way.
  In this paper we will show that a tame topological manifold is carried by a
tame homeomorphism to the interior of a compact piecewise linear manifolds
possibly with boundary and such a piecewise linear manifold possibly with
boundary is unique up to piecewise linear homeomorphisms in the sense that if
two manifolds are such PL manifolds possibly with boundary then they are the
same as piecewise linear manifolds. We modify this to Theorem 2 so that
argument of model theory works, and we prove it. We also consider the
differentiable case.

Geometric Topology