Approximating a topological section by a piecewise-linear section (Q2630668)

If for a surjective map \(f:X \to Y\) there is a map \(s:Y \to X\) such that \(f \circ s = id_Y\), then \(s\) is called a section of \(f\). The author proves that a surjective PL map \(f:X \to Y\) between compact polyhedra which has a topological section \(s:Y \to X\) has also PL section \(q:Y \to X\). Furthermore, \(q\) may be arbitrary close to \(s\) via homotopy through PL sections from \(s\) to \(q\). It is also pointed out that the same is true for locally compact polyhedra.