# Is the condition 'connected' necessary for differentiable structure?

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?

• 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 7 '19 at 5:34
• More importantly, is the condition of connectedness restrictive? i.e. are there manifolds with unconnected charts? – stressed out Jan 7 '19 at 6:15
• @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 7 '19 at 7:37
• @stressed Isn't it restrictive, as in my above comment? – zxcv Jan 7 '19 at 7:39
• @zxcv $\Bbb R$ is a differentiable manifold. – Angina Seng Jan 7 '19 at 7:46