Let $\omega\_1$ be smallest uncountable ordinal. I am trying to understand the possible "large" subspaces of $C[0,\omega\_1]$, namely those which are isomorphic to the whole space. Therefore I have the following question:


Does every subspace of $C[0,\omega\_1]$ isomorphic to $C[0,\omega\_1]$ contain a complemented copy isomorphic to itself? The only (complemented) examples that I can "construct by hand", excluding the finite-codimensional ones, are of the form


$\mbox{cl lin}(\mathbf{1}\_{[0,\gamma\_{\sigma}]}\colon \sigma\leq \omega\_1)$


where $(\gamma\_\sigma)\_{\sigma<\omega\_1}$ is increasing long sequence of limit ordinals and $\sigma\_{\omega\_1} = \omega\_1$ (note that the family $(\{\mathbf{1}\_{[0,\alpha]}\colon \alpha \leq \omega\_1\})$ forms the long Schauder basis for $C[0,\omega\_1]$).


Thank you,
T.



Subspaces isomorphic to $C[0, \omega_1]$
Yes, it can deduced from Lemma 1.2 combined with Proposition 2 of 



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> D. Alspach and Y. Benyamini, [Primariness of spaces of continuous functions on ordinals](http://link.springer.com/content/pdf/10.1007/BF02761606.pdf). *Israel J. Math*. **27** (1977), 64–92.
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> 
> 


Another proof can be found [here (Cor. 1.12)](http://arxiv.org/pdf/1303.0020v2.pdf).


