On light mappings without perfect fibers on compacta (Q1375717)

The setting is separable metric spaces. A continuous function \(f:X\to Y\) is light provided each fiber \(f^{-1}(y)\) is 0-dimensional. The paper includes results that ensure that, for a light mapping \(f:X\to Y\) between compacta, there is a nontrivial continuum \(C\subset X\) such that the restriction \(f|_C\) is one-to-one. One corollary of a key result (Theorem 1.1) is that, for a continuous mapping \(f:X\to Y\) between compacta with dimension of \(X\) at least 2, if each fiber \(f^{-1}(y)\) is countable, then there is a nontrivial continuum \(C\subset X\) such that the restriction \(f|_C\) is one-to-one. Combining Theorem 1.1 with a result due to H. Torunczyk leads to the corollary that, for a light mapping \(f:X\to Y\) between compacta with the dimension of \(X\) at least 3 and with \(Y\) finite dimensional, there is a nontrivial continuum \(C\subset X\) such that the restriction \(f|_C\) is one-to-one. An example is included that shows that the assumption that \(X\) is at least 3-dimensional is needed. Comparable results for strongly infinite-dimensional spaces are included.