Homeomorphism in the definition of a manifold

Question

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"?

Answer 1

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.

Answer 2

Note that $B(0,1) \simeq \mathbb{R}^n$. Can you answer your own question now?