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Let $X$ be a countable discrete topological space. Consider $X^{\mathbb{N}}$ endowed with the product topology.

How do you prove that $X^{\mathbb{N}}$ is homeomorphic to the sub-space of all irrational numbers?

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Take $X = \mathbb{N}$ and use continued fractions. –  t.b. Jun 17 '12 at 21:53
    
Look for "Baire Space" –  JBC Jun 17 '12 at 21:55
    
    
An alternativ proof can be found under theorem 1.1 at arxiv.org/pdf/math/9401202v1.pdf –  Michael Greinecker Jun 17 '12 at 22:18
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