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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