Strong paracompactness and multi-selections (Q972538)

The main theorem of this paper shows that a space \(X\) is strongly paracompact if and only if for each complete metric space \((Y,\rho)\) and each lower semi-continuous closed-valued multifunction \(\Phi:X\to Y\), there is a complete ultrametric space \((Z,d)\), an upper semi-continuous multifunction \(\psi:X\to Z\) and a uniformly continuous \(g:Z\to Y\) such that \(g\psi\) is a multi-selection for \(\Phi\) and \(\psi(\psi^{-1}(S))\) is totally \(\varepsilon\)-bounded when \(\varepsilon>0\) and \(S\subset Z\) is totally \(\varepsilon\)-bounded.