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?