The k-leader and mappings which change the dimensions of topological spaces

For a Tychonoff space X, let kX be the k-leader of X, dim X the covering dimension of X and Ind X the large inductive dimension of X. We prove: (1) For every n = 1,2,3,...,∞, there exists a regular Lindelof space X such that kX ix normal and dim X = Ind X = 0<n=dim kX = Ind lX. This shows that the natural map k:kX →X, which is a proper refinable map, can lower the dimensions. (2)If cf(c) > ω1, then for every n = 1,2,3,...,∞, there exists an open and closed, continuous map f from a normal space X onto a normal space Y such that dim f^-1 (y) = Ind f^-1 (y) = 0 for each y∊Y and dim X = n > 0 = dim Y = Ind Y.