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http://math.andrej.com/wp-content/uploads/2006/05/kleene-tree.pdf gives a good description of Cantor space, Weak Konig's Lemma, why it is classically true, why it is equivalent to compactness of Cantor space, why computable Cantor space is computably non compact, and the Kleene tree, which is an infinite computable tree without an infinite computable path.
Andre J's thesis talks about realizers and modest sets:
http://math.andrej.com/wp-content/uploads/2006/04/thesis.pdf
