On open and \(\alpha\)-covering mappings
Let \(0<\alpha<\infty\). A mapping between metric spaces \(f: X\to Y\) is \(\alpha\)-covering if the image of each open ball of radius \(r\) of the space \(X\) centered at a point \(p\) contains the ball of radius \(\alpha r\) of \(Y\) centered at \(f(p)\): \(f(O^X_r(p))\supset O^Y_{\alpha r}(f(p))\). In this paper, the author considers the following question: under what conditions can we make a given open mapping of compacta \(f:X\rightarrow Y\) 1-covering by replacing the metric of \(X\) or \(Y\) with a topologically equivalent metric? In particular the author obtains the following interesting results: (1) An open mapping \(f\) of a compactum \(X\) onto \(Y\) can be made 1-covering by replacing the metric on \(Y\) with an equivalent metric. (2) Among all such metrics on \(Y\), there exists a unique minimal metric. The author also gives some examples demonstrating the obtained results. In particular, an example is provided in which a mapping \(f\) cannot be made 1-covering by changing the metric of \(X\), and some examples are given showing that the condition that \(X\) is compact is substantial.
scientific article; zbMATH DE number 6908384