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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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    An alternativ proof can be found under theorem 1.1 at http://arxiv.org/pdf/math/9401202v1.pdf2012-06-17

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