definition of restricted \(C^\infty\) vectors bundle
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) vectors bundle.
- The reader knows a definition of immersed submanifold with boundary of \(C^\infty\) manifold with boundary.
- The reader admits the proposition that on any 2nd-countable topological space, any open cover of any subset has a countable subcover.
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary, any embedded submanifold with boundary of any embedded submanifold with boundary is an embedded submanifold with boundary of the manifold with boundary.
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary and any embedded submanifold with boundary, the inverse of the codomain restricted inclusion is \(C^\infty\).
- The reader admits the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction on any domain that contains the point is \(C^k\) at the point.
- The reader admits the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction or expansion on any codomain that contains the range is \(C^k\) at the point.
- The reader admits the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
- The reader admits the proposition that for any set, any to-be-atlas determines the canonical topology and the atlas.
- The reader admits the proposition that for any set and any 2 topology-atlas pairs, if and only if there is a common chart domains open cover and the transition for each common chart is a diffeomorphism, the pairs are the same.
Target Context
- The reader will have a definition of restricted \(C^\infty\) vectors bundle.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( M'\): \(\in \{\text{ the } d' \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\( E'\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( (E', M', \pi')\): \(\in \{\text{ the } C^\infty \text{ vectors bundles of rank } k\}\)
\( M\): \(\in \{\text{ the } d \text{ -dimensional immersed submanifolds with boundary of } M\}\)
\(\iota\): \(: M \to M'\), \(= \text{ the inclusion }\)
\( E\): \(= \pi'^{-1} (M) \subseteq E'\) with the topology and the atlas specified below
\( \pi\): \(= \pi \vert_{E}: E \to M\), \(\in \{\text{ the } C^\infty \text{ locally trivial surjections of rank } k\}\)
\(*(E, M, \pi)\):
//
Conditions:
For each \(m \in M\), take a trivializing open subset around \(m \in M'\), \(U'_m \subseteq M'\).
Take a chart around \(m \in M\), \((U_m \subseteq M, \phi_m)\), such that \(\iota (U_m) \subseteq U'_m\) and \(U_m\) is an embedded submanifold with boundary of \(M'\): as \(\iota\) is continuous, an open neighborhood of \(m\) on \(M\) can be taken to be mapped into \(U'_m\) under the inclusion and a chart domain can be taken inside the open neighborhood, while the chart domain can be an embedded submanifold with boundary of \(M'\) because any immersed submanifold with boundary is locally an embedded submanifold with boundary.
\(\{U_m \vert m \in M\}\) is an open cover of \(M\), and take any countable subcover, \(\{U_\beta \vert \beta \in B\}\), which is possible by the proposition that on any 2nd-countable topological space, any open cover of any subset has a countable subcover.
Denote the corresponding trivializing open subsets as \(\{U'_\beta \vert \beta \in B\}\) and take trivializations, \(\{\Phi'_\beta: \pi'^{-1} (U'_\beta) \to U'_\beta \times \mathbb{R}^k \vert \beta \in B\}\).
Let \(\Phi_\beta: \pi^{-1} (U_\beta) \to U_\beta \times \mathbb{R}^k := ({\iota'_\beta}^{-1}, id) \circ \Phi'_\beta \circ \tau_\beta\), where \(\tau_\beta: \pi^{-1} (U_\beta) \to \pi'^{-1} (U'_\beta)\) is the inclusion, \(\iota_\beta: U_\beta \to M'\) is the inclusion, and \(\iota'_\beta: U_\beta \to \iota_\beta (U_\beta) \subseteq M'\) is the codomain restriction of \(\iota_\beta\).
Let \(\lambda: \mathbb{R}^{d + k} \to \mathbb{R}^{d + k}, (r^1, ..., r^d, r^{d + 1}, ..., r^{d + k}) \mapsto (r^{d + 1}, ..., r^{d + k}, r^1, ..., r^d)\).
Let \(\widetilde{\phi_\beta}: \pi^{-1} (U_\beta) \to \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k}\), \(= \lambda \circ (\phi_\beta, id) \circ \Phi_\beta\).
Make the to-be-atlas for \(E\), \(\{(\pi^{-1} (U_\beta), \widetilde{\phi_\beta}) \vert \beta \in B\}\), determine the topology and the atlas of \(E\), by the proposition that for any set, any to-be-atlas determines the canonical topology and the atlas.
//
2: Note
Let us see that \(\{(\pi^{-1} (U_\beta), \widetilde{\phi_\beta}) \vert \beta \in B\}\) is indeed a to-be-atlas mentioned in the proposition that for any set, any to-be-atlas determines the canonical topology and the atlas.
\(\{\pi^{-1} (U_\beta) \vert \beta \in B\}\) is indeed a countable cover of \(E\).
\(\widetilde{\phi_\beta} (\pi^{-1} (U_\beta)) \subseteq \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k}\) is open, because \(\widetilde{\phi_\beta} (\pi^{-1} (U_\beta)) = \lambda \circ (\phi_\beta, id) \circ \Phi_\beta (\pi^{-1} (U_\beta)) = \lambda \circ (\phi_\beta, id) \circ ({\iota'_\beta}^{-1}, id) \circ \Phi'_\beta \circ \tau_\beta (\pi^{-1} (U_\beta)) = \lambda \circ (\phi_\beta, id) (U_\beta \times \mathbb{R}^k) = \lambda (\phi_\beta (U_\beta) \times \mathbb{R}^k) = \mathbb{R}^k \times \phi_\beta (U_\beta)\), which is open on \(\mathbb{R}^{d + k}\) or \(\mathbb{H}^{d + k}\).
\(\widetilde{\phi_\beta} (\pi^{-1} (U_\beta) \cap \pi^{-1} (U_{\beta'})) = \widetilde{\phi_\beta} (\pi^{-1} (U_\beta \cap U_{\beta'})) = \mathbb{R}^k \times \phi_\beta (U_\beta \cap U_{\beta'})\), which is open on \(\widetilde{\phi_\beta} (\pi^{-1} (U_\beta)) = \mathbb{R}^k \times \phi_\beta (U_\beta)\), because \(U_\beta \cap U_{\beta'}\) is open on \(U_\beta\), so, \(\phi_\beta (U_\beta \cap U_{\beta'})\) is open on \(\phi_\beta (U_\beta)\).
\(\widetilde{\phi_\beta}\) is obviously injective.
\(\widetilde{\phi_{\beta'}} \circ {\widetilde{\phi_\beta}}^{-1} \vert_{\widetilde{\phi_\beta} (\pi^{-1} (U_\beta) \cap \pi^{-1} (U_{\beta'}))} = \lambda \circ (\phi_{\beta'}, id) \circ \Phi_{\beta'} \circ {\Phi_\beta}^{-1} \circ (\phi_\beta, id)^{-1} \circ {\lambda}^{-1} \vert_{\mathbb{R}^k \times (\phi_\beta (U_\beta \cap U_{\beta'}))}\), because \(\widetilde{\phi_\beta} (\pi^{-1} (U_\beta) \cap \pi^{-1} (U_{\beta'})) = \widetilde{\phi_\beta} (\pi^{-1} (U_\beta \cap U_{\beta'})) = \lambda \circ (\phi_\beta, id) \circ \Phi_\beta (\pi^{-1} (U_\beta \cap U_{\beta'})) = \lambda \circ (\phi_\beta, id) ((U_\beta \cap U_{\beta'}) \times \mathbb{R}^k) = \lambda \circ (\phi_\beta (U_\beta \cap U_{\beta'}), \mathbb{R}^k) = \mathbb{R}^k \times (\phi_\beta (U_\beta \cap U_{\beta'}))\).
\(= \lambda \circ (\phi_{\beta'}, id) \circ ({\iota'_{\beta'}}^{-1}, id) \circ \Phi'_{\beta'} \circ \tau_{\beta'} \circ (({\iota'_\beta}^{-1}, id) \circ \Phi'_\beta \circ \tau_\beta)^{-1} \circ (\phi_\beta, id)^{-1} \circ {\lambda}^{-1} \vert_{\mathbb{R}^k \times (\phi_\beta (U_\beta \cap U_{\beta'}))} = \lambda \circ (\phi_{\beta'}, id) \circ ({\iota'_{\beta'}}^{-1}, id) \circ \Phi'_{\beta'} \circ \tau_{\beta'} \circ {\tau_\beta}^{-1} \circ {\Phi'_\beta}^{-1} \circ ({\iota'_\beta}^{-1}, id)^{-1} \circ (\phi_\beta, id)^{-1} \circ {\lambda}^{-1} \vert_{\mathbb{R}^k \times (\phi_\beta (U_\beta \cap U_{\beta'}))} = \lambda \circ (\phi_{\beta'}, id) \circ ({\iota'_{\beta'}}^{-1}, id) \circ \Phi'_{\beta'} \circ \tau_{\beta'} \circ {\tau'_\beta}^{-1} \circ {\Phi'_\beta}^{-1} \circ (\iota'_\beta, id) \circ (\phi_\beta, id)^{-1} \circ {\lambda}^{-1} \vert_{\mathbb{R}^k \times (\phi_\beta (U_\beta \cap U_{\beta'}))}\), where \(\tau'_\beta: \pi^{-1} (U_\beta) \to \tau_\beta (\pi^{-1} (U_\beta)) \subseteq \pi'^{-1} (U'_\beta)\) is the codomain restriction of \(\tau_\beta\).
\(= \lambda \circ (\phi_{\beta'}, id) \circ ({\iota'_{\beta'}}^{-1}, id) \circ \Phi'_{\beta'} \circ {\Phi'_\beta}^{-1} \circ (\iota'_\beta, id) \circ (\phi_\beta, id)^{-1} \circ {\lambda}^{-1} \vert_{\mathbb{R}^k \times (\phi_\beta (U_\beta \cap U_{\beta'}))}\), which is \(: \mathbb{R}^k \times (\phi_\beta (U_\beta \cap U_{\beta'})) \subseteq \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k} \to \phi_\beta (U_\beta \cap U_{\beta'}) \times \mathbb{R}^k \subseteq \mathbb{R}^d \times \mathbb{R}^k \text{ or } \mathbb{H}^d \times \mathbb{R}^k \to (U_\beta \cap U_{\beta'}) \times \mathbb{R}^k \subseteq U_\beta \times \mathbb{R}^k \to (U_\beta \cap U_{\beta'}) \times \mathbb{R}^k \subseteq M' \times \mathbb{R}^k \to \pi'^{-1} (U_\beta \cap U_{\beta'}) \subseteq \pi'^{-1} (U'_\beta \cap U'_{\beta'}) \to (U_\beta \cap U_{\beta'}) \times \mathbb{R}^k \subseteq M' \times \mathbb{R}^k \to (U_\beta \cap U_{\beta'}) \times \mathbb{R}^k \subseteq U_{\beta'} \times \mathbb{R}^k \to \phi_{\beta'} (U_\beta \cap U_{\beta'}) \times \mathbb{R}^k \subseteq \mathbb{R}^d \times \mathbb{R}^k \text{ or } \mathbb{H}^d \times \mathbb{R}^k \to \mathbb{R}^k \times \phi_{\beta'} (U_\beta \cap U_{\beta'}) \subseteq \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k}\).
The point is that \({\iota'_{\beta'}}^{-1}\) is \(C^\infty\), by the proposition that for any \(C^\infty\) manifold with boundary and any embedded submanifold with boundary, the inverse of the codomain restricted inclusion is \(C^\infty\). In fact, in order for that, we need \(U_\beta\) to be embedded in \(M'\).
\({\Phi'_\beta}^{-1}\) and \(\Phi'_{\beta'}\) are \(C^\infty\), because they are some trivializations for the established \(C^\infty\) vectors bundle, \((E', M', \pi')\).
Each constituent of \(\widetilde{\phi_{\beta'}} \circ {\widetilde{\phi_\beta}}^{-1} \vert_{\widetilde{\phi_\beta} (\pi^{-1} (U_\beta) \cap \pi^{-1} (U_{\beta'}))}\) is \(C^\infty\), by the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction on any domain that contains the point is \(C^k\) at the point and the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction or expansion on any codomain that contains the range is \(C^k\) at the point.
So, \(\widetilde{\phi_{\beta'}} \circ {\widetilde{\phi_\beta}}^{-1} \vert_{\widetilde{\phi_\beta} (\pi^{-1} (U_\beta) \cap \pi^{-1} (U_{\beta'}))}\) is a legitimate chain of \(C^\infty\) maps, and is \(C^\infty\), by the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
Also \(\widetilde{\phi_{\beta}} \circ {\widetilde{\phi_{\beta'}}}^{-1} \vert_{\widetilde{\phi_{\beta'}} (\pi^{-1} (U_{\beta'}) \cap \pi^{-1} (U_\beta))}\) is \(C^\infty\), by the symmetry.
Let us see that \(E\) is Hausdorff: see Note for the proposition that for any set, any to-be-atlas determines the canonical topology and the atlas. Let \(e, e' \in E\) be any such that \(e \neq e'\). When \(\pi (e) = \pi (e')\), there is a \(\pi^{-1} (U_\beta)\) such that \(e, e' \in \pi^{-1} (U_\beta)\). \(\pi^{-1} (U_\beta)\) is homeomorphic to \(\mathbb{R}^k \times \phi_\beta (U_\beta)\) and \(\widetilde{\phi_\beta} (e) \neq \widetilde{\phi_\beta} (e')\), and there are an open neighborhood of \(\widetilde{\phi_\beta} (e)\), \(U_{\widetilde{\phi_\beta} (e)} \subseteq \mathbb{R}^k \times \phi_\beta (U_\beta)\), and an open neighborhood of \(\widetilde{\phi_\beta} (e')\), \(U_{\widetilde{\phi_\beta} (e')} \subseteq \mathbb{R}^k \times \phi_\beta (U_\beta)\), such that \(U_{\widetilde{\phi_\beta} (e)} \cap U_{\widetilde{\phi_\beta} (e')} = \emptyset\), because \(\mathbb{R}^k \times \phi_\beta (U_\beta)\) is Hausdorff, because \(\mathbb{R}^{d + k}\) or \(\mathbb{H}^{d + k}\) is Hausdorff. Then, \({\widetilde{\phi_\beta}}^{-1} (U_{\widetilde{\phi_\beta} (e)})\) and \({\widetilde{\phi_\beta}}^{-1} (U_{\widetilde{\phi_\beta} (e')})\) are some open neighborhoods of \(e\) and \(e'\) respectively, and \({\widetilde{\phi_\beta}}^{-1} (U_{\widetilde{\phi_\beta} (e)}) \cap {\widetilde{\phi_\beta}}^{-1} (U_{\widetilde{\phi_\beta} (e')} = \emptyset\). When \(\pi (e) \neq \pi (e')\), there are an open neighborhood of \(\pi (e)\), \(U_{\pi (e)} \subseteq M\), and an open neighborhood of \(\pi (e')\), \(U_{\pi (e')} \subseteq M\), such that \(U_{\pi (e)} \cap U_{\pi (e')} = \emptyset\), because \(M\) is Hausdorff. \(U_{\pi (e)}\) and \(U_{\pi (e')}\) can be taken to be contained in \(U_\beta\) and \(U_{\beta'}\) respectively, where it may be or not be that \(U_\beta = U_{\beta'}\). \(\pi^{-1} (U_{\pi (e)}) \subseteq \pi^{-1} (U_\beta)\) is open because \(\widetilde{\phi_\beta} (\pi^{-1} (U_{\pi (e)})) = \mathbb{R}^k \times \phi_\beta (U_{\pi (e)}) \subseteq \mathbb{R}^k \times \phi_\beta (U_\beta)\), which is open. Likewise, \(\pi^{-1} (U_{\pi (e')}) \subseteq \pi^{-1} (U_\beta')\) is open. \(\pi^{-1} (U_{\pi (e)}) \cap \pi^{-1} (U_{\pi (e')}) = \emptyset\).
So, \(\{(\pi^{-1} (U_\beta), \widetilde{\phi_\beta}) \vert \beta \in B\}\) is indeed a to-be-atlas mentioned in the proposition that for any set, any to-be-atlas determines the canonical topology and the atlas.
\(\pi\) is \(C^\infty\): for each \(e \in E\), there is a \(\beta \in B\) such that \(e \in \pi^{-1} (U_\beta)\); take the charts, \((\pi^{-1} (U_\beta) \subseteq E, \widetilde{\phi_\beta})\) and \((U_\beta \subseteq M, \phi_\beta)\); \(\phi_\beta \circ \pi \circ {\widetilde{\phi_\beta}}^{-1}\) is \(: (v, \phi_\beta (p) \mapsto \phi_\beta (p))\), which is obviously \(C^\infty\).
Let us see that \(\Phi_\beta\) is a trivialization of rank \(k\).
As \(\widetilde{\phi_\beta} = \lambda \circ (\phi_\beta, id) \circ \Phi_\beta\), \(\Phi_\beta = (\phi_\beta, id)^{-1} \circ {\lambda}^{-1} \circ \widetilde{\phi_\beta} = ({\phi_\beta}^{-1}, id) \circ {\lambda}^{-1} \circ \widetilde{\phi_\beta}\), which is diffeomorphic, because \({\phi_\beta}^{-1}\) and \(\widetilde{\phi_\beta}\) are diffeomorphic.
For each \(m \in U_\beta\), \(\Phi_\beta \vert_{\pi^{-1} (m)}\) is 'vectors spaces - linear morphisms' isomorphic, because it equals \(\Phi'_\beta \vert_{\pi^{-1} (m)}\).
So, \((E, M, \pi)\) is indeed a \(C^\infty\) vectors bundle.
Let us see that the topology and the atlas are uniquely defined: the procedure above ostensibly depends on the choices of \(\{(U_\beta, \phi_\beta) \vert \beta \in B\}\) and \(\{\Phi'_\beta \vert \beta \in B\}\).
Let another choices be \(\{(\overline{U_\gamma}, \overline{\phi_\gamma)} \vert \gamma \in \overline{B}\}\) and \(\{\overline{\Phi'_\gamma} \vert \gamma \in \overline{B}\}\).
Then, we have \(\overline{\Phi_\gamma} = ({\iota'_\gamma}^{-1}, id) \circ \overline{\Phi'_\gamma} \circ \tau_\gamma\) and \(\widetilde{\overline{\phi_\gamma}}: \pi^{-1} (\overline{U_\gamma}) \to \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k} = \lambda \circ (\overline{\phi_\gamma}, id) \circ \overline{\Phi_\gamma}\).
We are going to apply the proposition that for any set and any 2 topology-atlas pairs, if and only if there is a common chart domains open cover and the transition for each common chart is a diffeomorphism, the pairs are the same.
\(\{\pi^{-1} (U_\beta) \vert \beta \in B)\}\) is a chart domains open cover for the former topology-atlas pair and \(\{\pi^{-1} (\overline{U_\gamma}) \vert \gamma \in \overline{B})\}\) is a chart domains open cover for the latter topology-atlas pair. \(\{\pi^{-1} (U_\beta) \cap \pi^{-1} (\overline{U_\gamma}) \vert (\beta, \gamma) \in B \times \overline{B})\}\) is a countable common chart domains open cover, because \(\pi^{-1} (U_\beta) \cap \pi^{-1} (\overline{U_\gamma}) = \pi^{-1} (U_\beta \cap \overline{U_\gamma}) = {\Phi_\beta}^{-1} ((U_\beta \cap \overline{U_\gamma}) \times \mathbb{R}^k)\), which is open on \(\pi^{-1} (U_\beta)\), and likewise, it is open on \(\pi^{-1} (\overline{U_\gamma})\).
\(\widetilde{\overline{\phi_\gamma}} \circ {\widetilde{\phi_\beta}}^{-1} \vert_{\widetilde{\phi_\beta} (\pi^{-1} (U_\beta) \cap \pi^{-1} (\overline{U_\gamma}))} = \lambda \circ (\overline{\phi_\gamma}, id) \circ \overline{\Phi_\gamma} \circ {\Phi_\beta}^{-1} \circ (\phi_\beta, id)^{-1} \circ {\lambda}^{-1} \vert_{\mathbb{R}^k \times \phi_\beta (U_\beta \cap \overline{U_\gamma})}\), because \(\widetilde{\phi_\beta} (\pi^{-1} (U_\beta) \cap \pi^{-1} (\overline{U_\gamma})) = \widetilde{\phi_\beta} (\pi^{-1} (U_\beta \cap \overline{U_\gamma})) = \mathbb{R}^k \times \phi_\beta (U_\beta \cap \overline{U_\gamma})\).
\(= \lambda \circ (\overline{\phi_\gamma}, id) \circ ({\iota'_\gamma}^{-1}, id) \circ \overline{\Phi'_\gamma} \circ \tau_\gamma \circ (({\iota'_\beta}^{-1}, id) \circ \Phi'_\beta \circ \tau_\beta)^{-1} \circ ({\phi_\beta}^{-1}, id) \circ {\lambda}^{-1} = \lambda \circ (\overline{\phi_\gamma}, id) \circ ({\iota'_\gamma}^{-1}, id) \circ \overline{\Phi'_\gamma} \circ \tau_\gamma \circ {\tau_\beta}^{-1} \circ {\Phi'_\beta}^{-1} \circ ({\iota'_\beta}^{-1}, id)^{-1} \circ ({\phi_\beta}^{-1}, id) \circ {\lambda}^{-1} = \lambda \circ (\overline{\phi_\gamma}, id) \circ ({\iota'_\gamma}^{-1}, id) \circ \overline{\Phi'_\gamma} \circ {\Phi'_\beta}^{-1} \circ ({\iota'_\beta}^{-1}, id)^{-1} \circ ({\phi_\beta}^{-1}, id) \circ {\lambda}^{-1}\).
That is \(: \mathbb{R}^k \times \phi_\beta (U_\beta \cap \overline{U_\gamma}) \subseteq \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k} \to \phi_\beta (U_\beta \cap \overline{U_\gamma}) \times \mathbb{R}^k \subseteq \mathbb{R}^d \times \mathbb{R}^k \text{ or } \mathbb{H}^d \times \mathbb{R}^k \to (U_\beta \cap \overline{U_\gamma}) \times \mathbb{R}^k \subseteq U_\beta \times \mathbb{R}^k \to (U_\beta \cap \overline{U_\gamma}) \times \mathbb{R}^k \subseteq M' \times \mathbb{R}^k \to \pi^{-1} (U_\beta \cap \overline{U_\gamma}) \subseteq \pi^{-1} (U'_\beta \cap \overline{U'_\gamma}) \to (U_\beta \cap \overline{U_\gamma}) \times \mathbb{R}^k \subseteq M' \times \mathbb{R}^k \to (U_\beta \cap \overline{U_\gamma}) \times \mathbb{R}^k \subseteq \overline{U_\gamma} \times \mathbb{R}^k \to \overline{\phi_\gamma} (U_\beta \cap \overline{U_\gamma}) \times \mathbb{R}^k \subseteq \mathbb{R}^d \times \mathbb{R}^k \text{ or } \mathbb{H}^d \times \mathbb{R}^k \to \mathbb{R}^k \times \overline{\phi_\gamma} (U_\beta \cap \overline{U_\gamma}) \subseteq \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k}\), which is a legitimate chain of \(C^\infty\) maps.
So, \(\widetilde{\overline{\phi_\gamma}} \circ {\widetilde{\phi_\beta}}^{-1} \vert_{\widetilde{\phi_\beta} (\pi^{-1} (U_\beta) \cap \pi^{-1} (\overline{U_\gamma}))}\) is \(C^\infty\), by the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
Also, \(\widetilde{\phi_\beta} \circ {\widetilde{\overline{\phi_\gamma}}}^{-1} \vert_{\widetilde{\overline{\phi_\gamma}} (\pi^{-1} (\overline{U_\gamma}) \cap \pi^{-1} (U_\beta))}\) is \(C^\infty\), by the symmetry.
So, \(\widetilde{\overline{\phi_\gamma}} \circ {\widetilde{\phi_\beta}}^{-1} \vert_{\widetilde{\phi_\beta} (\pi^{-1} (U_\beta) \cap \pi^{-1} (\overline{U_\gamma}))}\) is diffeomorphic.
So, by the proposition that for any set and any 2 topology-atlas pairs, if and only if there is a common chart domains open cover and the transition for each common chart is a diffeomorphism, the pairs are the same, the 2 topology-atlas pairs are the same, which means that the topology and the atlas are uniquely determined by the specification.
Let us see that \(E\) is an immersed submanifold with boundary of \(E'\).
Let \(\tau: E \to E'\) be the inclusion.
Although \(U'_\beta\) was chosen to be just a trivializing open subset, \(U'_\beta\) can be a chart trivializing open subset, which we do now.
For each \(e \in E\), let us choose the charts, \((\pi^{-1} (U_\beta) \subseteq E, \widetilde{\phi_\beta})\) and \((\pi'^{-1} (U'_\beta) \subseteq E', \widetilde{\phi'_\beta})\), such that \(e \in \pi^{-1} (U_\beta)\), which inevitably implies that \(\tau (e) \in \pi'^{-1} (U'_\beta)\) and \(\tau (\pi^{-1} (U_\beta)) \subseteq \pi'^{-1} (U'_\beta)\).
\(\widetilde{\phi'_\beta} \circ \tau \circ {\widetilde{\phi_\beta}}^{-1}\) is \(: (v, \phi_\beta (p)) \mapsto (v, \phi'_\beta (p)) = (id, \phi'_\beta \circ {\phi_\beta}^{-1})\), which implies that \(\tau\) is an injective \(C^\infty\) immersion, because \(\phi'_\beta \circ {\phi_\beta}^{-1}\) satisfies the characteristics that makes \(\iota: M \to M'\) be an injective \(C^\infty\) immersion.
