Functors of iterated superextension (Q912424)

Given a compact metric space X, denote by \(X\to^{\eta_ X}\lambda X\) its superextension. If \(\alpha\) : \(N\to N\) is a map, \(O\leq \alpha (n)\leq n\) (n\(\in N)\), we have the maps \(\lambda^{\alpha (n)}\) \((\eta_{\lambda^{n-\alpha (n)}X}):\) \(\lambda^ nX\to \lambda^{n+1}X\). These maps constitute a chain \(X\to \lambda X\to \lambda^ 2X\to...\to \lambda^ nX\to..\). of isometries. Its colimit is a metric space \(F^{\alpha}X\); denote by \(\tilde F^{\alpha}X\) its completion. Main result: The pair \(F^{\alpha}X\), \(\tilde F^{\alpha}X\) is topologically homeomorphic to \(\Sigma\), \(\ell_ 2\) where \(\Sigma\) is the linear span of the Hilbert cube Q embedded in \(\ell_ 2\) in the standard way.