Let $X$ be a set and $(U_i, \psi_i)_{i \in I}$ be an [[Atlas|atlas]] of class $C^p$ with $p \ge 1$ and $\psi_i: U_i \to E_i$. > [!theorem] > > If $U_i \cap U_j \ne \emptyset$, then $E_i \iso E_j$. > > *Proof*. Since the intersection is non-empty, for suitable domain and target, both the maps $F = \varphi_i \circ \varphi_j^{-1}$ and $G = \varphi_j \circ \varphi_i^{-1}$ are $C^1$ isomorphisms. In particular, $F \circ G: E_j \to E_j$ is the identity, and by the chain rule, > $ > D(F \circ G)(x) = DF(G(x)) \circ DG(x) = I > $ > Therefore $DG(x): E_i \to E_j$ is a [[Space of Toplinear Isomorphisms|toplinear isomorphism]]. > [!theorem] > > Let $E$ be a [[Banach Space|Banach space]], then the collection > $ > M_E = \bracs{x \in X: \exists i \in I: x \in U_i, E_i \iso E} > $ > is both [[Open Set|open]] and [[Closed Set|closed]] with respect to the [[Topology on Manifold|topology]] induced by $(U_i, \psi_i)_{i \in I}$. > > *Proof*. Let $x \in M_E$ and let $i_x$ such that $x \in U_{i_x}$ and $E_{i_x} \iso E$, $M_E = \bigcup_{x \in M_E}U_{i_x}$ is a union of open sets. > > Let $x \in \ol{M_E}$, then for any open set $U$, $U \cap M_E \ne \emptyset$. Let $i \in I$ such that $x \in U_i$, then $U_i \cap M_E \ne \emptyset$, hence $E_i \iso E$, and $x \in M_E$, making $M_E$ closed. > [!theorem] > > Let $C \subset X$ be a [[Connected|connected]] set with respect to the [[Topology on Manifold|final topology]], then for any $x, y \in C$, the target of their charts are isomorphic. > > *Proof*. Let $\psi_i$ be a chart for $X$, $E$ be its target, and $M_E$ as above. > > Suppose that there exists $y \in C$ such that the target of every chart of $y$ is not isomorphic to $E$, that is, $y \not\in M_E$. In this case, since $M_E$ is both open and closed, $M_E$ and $M_E^c$ are disjoint open sets that both intersect $C$. This contradicts with the fact that $C$ is connected. Therefore $C \subset M_E$.