In John Lee's Book "Introduction to Smooth Manifolds" on page 17, Example 1.12, the author states that the smooth structure on any zero - dimensional manifold is unique. That confuses me, suppose for example the $M = \{ p \}$ is a point. Then I can for example give this manifold a smooth structure by taking $(U, \psi)$ to be given by \begin{equation} U = \{ p \}, \quad \psi(p) = 1 \end{equation} or, I could also give it the structure \begin{equation} U = \{ p \}, \quad \psi(p) = 2 \end{equation} From what I understand, these are different smooth structures. What am I missing ?
Uniqueness of smooth structure on a zero-dimensional smooth manifold
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differential-geometry
manifolds
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1You're off by one dimension. the charts should map to $R^0$, not $R^1$. – 2015-12-13
1 Answers
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As Dyland Moreland points out $\mathbb{R}^0=\{p\}$. Thus, there is for each discrete space $X$ a unique zero-dimensional manifold structure with the charts $\psi_x:\{x\}\to\mathbb{R}^0$ being the unique such maps.
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1Part of the definition of being a smooth manifold is that the transition maps are smooth. In what sense are these transition maps smooth? In what sense is the (only) map $\Bbb R^0\to\Bbb R^0$ smooth? – 2017-11-11