Tietze extensions and continuous selections for metric projections (Q807896)

Let T be a locally compact Hausdorff topological space and S a compact subset of T. For \(g\in C(S)\) denote by \(E(g)=\{g\in C_ 0(T):\) \(f|_ S=g\), \(\| f\| =\| g\|_ S\}\) the set of all Tietze extensions of the function g. Denote by \(P_ M\) the metric projection of \(C_ 0(T)\) onto the closed ideal \(M=\{f\in C_ 0(T):\) \(f|_ M=0\}\). The aim of this paper is to show that the properties of the multivalued applications E and \(P_ M\) are closely related. For instance, \(P_ M(f)=f-E(f|_ S)\), for all \(f\in C_ 0(T)\) and E admits a linear (respectively Lipschitz) selection if and only if \(P_ M\) does. The paper contains also other results concerning the applications E and \(P_ M\)- for instance, they are both Lipschitz with respect to the Hausdorff metric.