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In Calculus on Manifolds Spivak gives the following definition of a manifold: enter image description here

But $\Bbb{R}^k \times \{0\}$ is a set of $(k+1)$-tuples and $V$ is a set of $n$-tuples, so $$V \cap (\Bbb{R}^k \times \{0\}) = \emptyset $$ if $k+1 \neq n$. This doesn't add up with what Spivak states it's equal to.

Shouldn't it be $\Bbb{R}^k \times \{0\}^{n - k}$?

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Here in the expression $\mathbb R^k \times \{0\}$ the symbol $0$ denotes the zero vector in $\mathbb R^{n-k}$. Often the same symbol $0$ is used to mean different things. The description $\{ y \in V \mid y^{k+1} = \cdots = y^n = 0 \}$ might be more clear.

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  • $\begingroup$ "The vector $(0,\dotsc,0)$ will usually be denoted simply $0$." is stated on p.3 of the book. $\endgroup$
    – Thorgott
    Feb 25 at 13:29
  • $\begingroup$ Thank you. But I find this to be a pretty sloppy definition since tuples of different dimensions are not the same and there is no way to denote the dimension in this notation $\endgroup$
    – Sgg8
    Feb 25 at 13:41

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