On closed mappings of $\sigma$-compact spaces and dimension

Abstract:

A remainder of the Hilbert space $l_{2}$ is a space homeomorphic to $Z \setminus l_{2}$, where $Z$ is a metrizable compact extension of $l_{2}$, with $l_{2}$ dense in $Z$. We prove that for any remainder $K$ of $l_{2}$, every non-one-point closed image of $K$ contains either a compact set with no transfinite dimension or compact sets of arbitrarily high inductive transfinite dimension $\mathrm {ind}$. We shall also construct 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 inductive dimension $\mathrm {ind}$ (this provides a rather strong negative solution of a problem in [Dissertationes Math. (Rozprawy Mat.) 216 (1983), pp. 1–41]).