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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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