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I'm studying differentiable manifolds with the book Warner. "Foundations of Differentiable Manifolds and Lie Groups." It defines differentiable structure as follows but I think maybe the connectedness condition is included mistakenly.

1.3 Definitions $\;$ A locally Euclidean space $M$ of dimension $d$ is a Hausdorff topological space $M$ for which each point has a neighborhood homeomorphic to an open subset of Euclidean space $\mathbb R^d$. If $\varphi$ is a homeomorphism of a connected open set $U\subset M$ onto an open subset of $\mathbb R^d$, $\varphi$ is called a coordinate map, the functions $x_i = r_i \circ \varphi$ are called the coordinate functions, and the pair $(U, \varphi)$ (somethimes denoted by $(U, x_1, \ldots, x_d)$) is called a coordinate system. (The rest omitted because it is irrelavent to my question.)

1.4 Definitions $\;$ A differentiable structure $\mathcal F$ of class $C^k$ $(1\leq k\leq \infty)$ on a locally Euclidean space $M$ is a collection of coordinate systems $\{(U_\alpha, \varphi_\alpha): \alpha\in A\}$ satisfying the following three properties:
(a) $\bigcup_{\alpha\in A}U_\alpha = M$.
(b) $\varphi_\alpha\circ\varphi_\beta^{-1}$ is $C^k$ for all $\alpha,\beta\in A$.
(c) The collection $\mathcal F$ is maximal with respect to (b); that is, if $(U,\varphi)$ is a coordinate system such that $\varphi\circ\varphi_\alpha^{-1}$ and $\varphi_\alpha\circ\varphi^{-1}$ are $C^k$ for all $\alpha\in A$, then $(U,\varphi)\in\mathcal F$.

And here is the definition of differentiable manifold in the book.

A $d$-dimensional differentiable manifold of class $C^k$ is a pair $(M, \mathcal F)$ consisting of a $d$-dimensional, second countable, locally Euclidean space $M$ together with a differentiable structure $\mathcal F$ of class $C^k$.

My question is: Is it standard to put connectedness condition in definition 1.3? Becuase the book seems to ignore the connectedness condition in the texts that follows the above definitions (I'm not sure though, because I only read very little). For example, see the following.

An open subset $U$ of a differentiable manifold $(M,\mathcal F_M)$ is itself a differentiable manifold with the differentiable structure $\mathcal F_U = \{(U_\alpha\cap U,\varphi_\alpha|U_\alpha\cap U):(U_\alpha,\varphi_\alpha)\in\mathcal F_M)$.

Isn't this incorrect if there is a connectedness condition in definition 1.3?

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zxcv
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  • The condition of connectedness is redundant for the definition of manifold, but perhaps Warner finds it convenient for his "coordinate maps" to have connected image. – Angina Seng Jan 07 '19 at 05:34
  • More importantly, is the condition of connectedness restrictive? i.e. are there manifolds with unconnected charts? – stressed out Jan 07 '19 at 06:15
  • @Lord Why is it redundant? If there were no connectedness condition, can't $(\mathbb R, \mathcal F)$ be a differentiable manifold, where $\mathcal F$ is a differentiable structure containing $\varphi_1: \mathbb R-{0}\to\mathbb R-{0}, x\mapsto x$ and $\varphi_2: \mathbb R\to\mathbb R, x\mapsto x$? Having a connectedness condition will not allow $\varphi_1$ to be in $\mathcal F$ – zxcv Jan 07 '19 at 07:37
  • @stressed Isn't it restrictive, as in my above comment? – zxcv Jan 07 '19 at 07:39
  • @zxcv $\Bbb R$ is a differentiable manifold. – Angina Seng Jan 07 '19 at 07:46
  • @zxcv Yes, I think your example is a good one. It seems that one can get different maximal atlases in this way. However, I have a feeling that the two manifolds with the same base space and different maximal atlases obtained in this way (one allowing connected sets and the other not) will be diffeomorphic at the end. I guess that since our space is locally Euclidean, every open set will have a connected open subset. But I'm not sure. I'm just thinking out loud. – stressed out Jan 07 '19 at 07:47

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