On spaces splittable over the line and over \(R^ n\) (Q1898560)

A space \(X\) is splittable (or cleavable) over a space \(Y\) if for every \(A \subset X\) there exists a continuous mapping \(f : X \to Y\) such that \(f^{-1} f(A) = A\). The paper contains some results (usually without proofs) on splittability over reals \(\mathbb{R}\) and over \(\mathbb{R}^n\). A compactum (an infinite continuum) is splittable over \(\mathbb{R}\) if and only if it is homeomorphic to a subspace of \(\mathbb{R}\) (to a segment of \(\mathbb{R}\)). Every one-dimensional polyhedron is splittable over \(\mathbb{R}^2\). The sphere \(S^n\) and the spaces \(\mathbb{R}^m\), \(m > n \geq 1\), are not splittable over \(R^n\).