Many texts will define a manifold as "a second-countable Hausdorff space that is locally homeomorphic to Euclidean space". By definition of homeomorphism, shouldn't this really and officially read as "locally homeomorphic to a subset of Euclidean space"?
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I think the word locally is meant to apply to both the manifold and Euclidean space so it's implied to some degree. – Cameron L. Williams May 27 '15 at 19:35
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Locally homeomorphic to a subset of Euclidean space does not a manifold make. If you were to say an open subset, then it is a manifold. But it is equivalent to say all of Euclidean space. – Matt Samuel May 27 '15 at 19:38
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I don't understand the significance of the phrase "by definition of homeomorphism." Can you elaborate? – Qiaochu Yuan May 27 '15 at 19:40
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Definitely not: "locally homeomorphic to an open subset of Euclidean space" would be equivalent to the stated (and standard) definition, but Euclidean $n$-space for any $n > 0$ has subsets that are not manifolds, e.g., $\{0\} \cup \{1/n : 0 < n \in \mathbb{N}\} \subseteq \mathbb{R}$ or the union of the $x$-axis and the $y$-axis in $\mathbb{R}^2$. Any such subset would be a manifold according to your proposed alternative definition.
Rob Arthan
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