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A basis of the space of continuous function of countable ordinals $C({\alpha}) = C [0, {\alpha}]$, which consist of characteristics functions of clopen subsets of $C({\alpha})$, in some order. But can some one help me to know some details about the cases that how to pick a basis element in successor and in limit ordinal cases, with a example for the space say $C [0, \omega^3]$ or $C [0,\omega^{\omega}]$ .

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When you say 'In some order', do you have a specific order in mind? Also, please edit this to include an actual question, and some deatails, as per mathoverflow.net/howtoask –  David Roberts Nov 30 '12 at 2:21
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The best way to understand this consists in studying Bourgain's representation of the spaces $C(\alpha)$ as tree spaces; this is from Section 2 of Bourgain's paper The Szlenk index and operators on $C(K)$ spaces. Bourgain doesn't mention bases there, however an exposition of Bourgain's ideas, along with an observation of Odell about bases of separable $C(\alpha)$ spaces as per your question, can be found in Section 4.B of Rosenthal's survey article on $C(K)$ spaces in Volume 2 of the Handbook of the Geometry of Banach Spaces (in particular, see page 1583 onwards). –  Philip Brooker Nov 30 '12 at 11:38

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