On extensions of the domain invariance theorem (Q1399163)

Let \(X\) be a compact subset of \(\mathbb{R}^n\) and suppose \(x_0\) is in the interior of \(X\). The main result of this well-written paper states that if \(f: X\to\mathbb{R}^n\) is a map such that, for every point \(x\) on \(\partial X\), the boundary of \(X\), the diameter of \(f^{-1}(f(x))\) is less than half the distance from \(x_0\) to \(\partial X\), then \(f(x_0)\) is in the interior of \(f(X)\). Consequences of this result include the Brouwer invariance of the domain theorem as well as theorems of Borsuk and Sitnikov, concerning maps to \(\mathbb{R}^n\) from \(\mathbb{R}^n\) and an open subset \(U\) of \(\mathbb{R}^n\), respectively, in which conditions on the diameter of \(f^{-1}(f(x))\) imply that the map has open image.