On linear continuous open surjections of the spaces \(C_ p(X)\) (Q1375166)

Let \(X,Y\) be compact metric spaces and \(C_p(X)\) denote the linear space of continuous real valued functions on \(X\) with the topology of pointwise convergence. Arkhangel'skij posed the following questions: Suppose there is a continuous linear mapping \(L\) from \(C_p(X)\) onto \(C_p(Y)\). (i) Does it follow that \(\dim Y\leq\dim X\)? (ii) Does \(\dim Y\leq\dim X\) follow if we also assume that \(L\) is open? \textit{A. Leiderman, S. Morris} and \textit{V. Pestov} show, in a paper to appear in J. London Math. Soc., that the answer to (i) is negative. In the present paper, the authors show that the answer to (ii) is negative by constructing, for any finite-dimensional \(Y\), a 2-dimensional \(X\) that admits an open \(L\) from \(C_p(X)\) onto \(C_p(Y)\). They obtain other results relating the dimensions of \(X\), \(Y\) to the existence of an open \(L\) and show, among other things, that when \(X,Y\) are closed intervals, there exists an \(L\) which is not open.