Rimcompactness and similar properties in preimages (Q1201200)

The following is proved. Theorem 1: Suppose \(f: X\to Y\) is rimcompact and, for \(y\in Y\), \(f^{-1}(y)\) is rimcompact and has 0-dimensional boundary in \(X\). If \(Y\) is rimcompact then \(X\) is rimcompact. --- Theorem 2: Suppose \(f: X\to Y\) is rimperfect and, for \(y\in Y\), \(f^{-1}(y)\) is a TDI space (i.e. \(C(\beta f^{-1}(y))\) is an upper semicontinuous decomposition of \(\beta f^{-1}(y)\) consisting of compact sets with 0- dimensional boundary in \(X\)). If \(Y\) is a TDI space then \(X\) is a TDI space. Examples show that the hypotheses are not superfluous.