I'm studying Theorem 6.1.6 in Cohn's Measure Theory, which states that the product space $ \mathbb{N}^{\infty} $ is homeomorphic to the space of irrational numbers in $ (0,1) $, equipped with the usual topology.
In the proof, the author defines the map $$ h((n_i)) := \sum_{k=1}^{\infty} 2^{-n_1 - \cdots - n_k},(n_i) \in\mathbb{N}^\infty $$ and claims that it is a homeomorphism from $ \mathbb{N}^\infty $ onto $ [0,1] \setminus M $ where $ M $ denotes the countable set of binary rational numbers in $ [0,1] $.
However, I'm having difficulty understanding how this claim is fully justified. In particular, some sequences seem to yield binary rational values (e.g., $ n_i = 1 $ for all $i $ gives $h = 1 $), so it's not immediately clear how the image avoids $ M $, or how injectivity and continuity are preserved under this mapping.
Any insight or clarification would be greatly appreciated.
