Infinite internal direct sums of subspaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T19:46:24Z http://mathoverflow.net/feeds/question/66451 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66451/infinite-internal-direct-sums-of-subspaces Infinite internal direct sums of subspaces Wiktor Jaszak 2011-05-30T13:55:32Z 2011-05-31T09:27:51Z <p>Given a compact Hausdorff space $K$ such that $C(K)$ is of density $\omega_1$. Suppose that every copy of $c_0(\omega_1)$ in $C(K)$ is complemented. Let ${Y_\alpha\colon\alpha&lt;\omega_1}$ be a family of copies of $c_0(\omega_1)$ in $C(K)$ such that</p> <p>$Y_\alpha\cap Y_\beta={0}$ for $\alpha\neq \beta$</p> <p>Can we describe somehow the closure $Y$ of the subspace</p> <p>$\sum_{\alpha&lt;\omega_1}Y_\alpha$?</p> <p>May it be isomorphic to $c_0(\omega_1)$ in some particular cases?</p>