Fiber \(Z\)-sets and fiber rejection homeomorphisms (Q1316866)

The author establishes fibered versions of two well-known results. These are the contraction mapping theorem and the result that \(X-K\) is homeomorphic to \(X\) whenever \(K\) is a complete subset of the non-complete normed linear space \(X\). These results are then put to use to obtain some corollaries related to the problem of finding a fiber preserving homeomorphism (close to the identity) of \(E-K\) onto \(E\) when \(p: E\to B\) is a bundle and \(K\subset E\) is a fibered \(Z\)-set For example, this problem has an affirmative solution in the case of the trivial \(l_2\)- bundle over a metrizable space \(B\).