Small Valdivia compact spaces (Q2502977)

It is shown that the class of Valdivia compact spaces is closed under limits of inverse systems \(\{X_\alpha, r_\alpha^\beta\}_{\alpha,\beta\in\kappa}\) such that \(X_0\) is Valdivia compact and all \(r_\alpha^{\alpha+1}\) are simple retractions. That result especially applies to metric compact spaces \(X_\alpha\) and retractions \(r_\alpha^{\alpha+1}\). As a corollary, a compact space \(X\) with \(w(X)\leq\omega_1\) is Valdivia iff it is a limit of an inverse system of metric spaces with retractions as bonding maps. Another consequence gives a preservation of Valdivia compacts by retractions onto spaces with \(w(X)\leq\omega_1\) or by open surjections onto zero-dimensional spaces with \(w(X)\leq\omega_1\). Using functorial properties of the construction of certain inverse systems, the authors prove that the class of Valdivia compacts is stable under continuous weight preserving covariant functors in compact spaces.