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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 $\{Y_\alpha\colon\alpha<\omega_1\}$ be a family of copies of $c_0(\omega_1)$ in $C(K)$ such that

$Y_\alpha\cap Y_\beta=\{0\}$ for $\alpha\neq \beta$

Can we describe somehow the closure $Y$ of the subspace

$\sum_{\alpha<\omega_1}Y_\alpha$?

May it be isomorphic to $c_0(\omega_1)$ in some particular cases?

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I added the Functional Analysis tag; apologies if you think it's not appropriate. –  Zen Harper May 31 '11 at 0:22
2  
No problem with the tags, but the question is too vague. –  Bill Johnson May 31 '11 at 0:51
    
I am wondering what is the answer if we take $K=[0,\omega_1]$. Can $Y$ (following the notation) be isomorphic to the whole space for some $Y_\alpha$s? –  Tomek Kania May 31 '11 at 9:28
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Tomek, let $\psi$ be a bijection of $\omega_1\times\omega_1$ onto the countable successor ordinals, $\phi$ a bijection of $\omega_1$ onto the countable limit ordinals, for $0<\alpha<\omega_1$ let $Y_\alpha$ be the closed span of $\{\chi_{\{\psi((\alpha,\eta))\}}\mid \eta<\omega_1\}\cup\{\chi_{(\alpha',\phi(\alpha)]}\}$, where $\alpha'=\min\{\mu\mid\exists\nu\mbox{ such that }\phi(\alpha)=\mu+\omega^\nu\}$, and let $Y_0$ be the closed span of $\{\chi_{\{\psi((0,\eta))\}}\mid \eta<\omega_1\}\cup\{\chi_{\{0\}}\}\cup\{\chi_{[0,\omega_1]}\}$. Then each $Y_\alpha$ is isomorphic to $c_0(\omega_1)$... –  Philip Brooker May 31 '11 at 15:39
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and we have $Y_\alpha\cap Y_\beta=\{0\}$ when $\alpha\neq\beta$. Moreover, the closed span of all the $Y_\alpha$s is all of $C([0,\omega_1])$. Well, I am pretty sure all that is true - let me know if I have made a mistake! I haven't checked it closely but am pretty sure it is correct. –  Philip Brooker May 31 '11 at 15:43

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