  We prove that if K is a remainder of the Hilbert space (i.e., K is the
complement of the Hilbert space in its metrizable compactification) then every
non-one-point closed image of K either contains a compact set with no
transfinite dimension or contains compact sets of arbitrarily high inductive
transfinite dimension ind. We construct also for each natural n a sigma-compact
metrizable n-dimensional space whose image under any non-constant closed map
has dimension at least n, and analogous examples for the transfinite dimension
ind.
