Reduction to general position of a mapping of a one-dimensional polyhedron, depending continuously on a parameter

This paper is devoted to a proof of the fact that by refining the triangulation of a one-dimensional polyhedron, one can approximate a given mapping of that polyhedron into $\mathbf R^k$ by a piecewise linear mapping having no more than a zero-dimensional violation of general position; and that all this can be carried out continuously with respect to a parameter running through a strongly paracompact space. Spaces of triangulations of one-dimensional simplexes are also investigated, and the structure of spaces of semilinear mappings of a one-dimensional polyhedron into Euclidean space is considered.
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