<The previous article in this series | The table of contents of this series |
description/proof of that for \(C^\infty\) vectors bundle and section from subset of base space \(C^k\) at point where \(0 \lt k\), there is \(C^k\) extension on open-neighborhood-of-point domain
Topics
About:
\(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
-
The reader will have a description and a proof of the proposition that for any \(C^\infty\) vectors bundle and any section from any subset of the base space \(C^k\) at any point where \(0 \lt k\), there is a \(C^k\) extension on an open-neighborhood-of-point domain.
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:
\((E, M, \pi)\): \(\in \{\text{ the } C^\infty \text{ vectors bundles of rank } k\}\)
\(S\): \(\subseteq M\)
\(p\): \(\in S\)
\(s\): \(: S \to \pi^{-1} (S) \subseteq E\), \(\in \{\text{ the sections of } \pi \vert_{\pi^{-1} (S)}\}\), \(C^l\) at \(p\), where \(0 \lt l\)
//
Statements:
\(\exists V_p \subseteq M \in \{\text{ the open neighborhoods of } p\}, \exists s': V_p \to \pi^{-1} (V_p) \in \{\text{ the sections of } \pi \vert_{\pi^{-1} (V_p)}\} (s \vert_{S \cap V_p} = s' \vert_{S \cap V_p} \land s' \in \{\text{ the } C^l \text{ maps }\})\)
//
2: Note
Compare with the proposition that for any map from any subset of any \(C^\infty\) manifold with boundary into any subset of any \(C^\infty\) manifold \(C^k\) at any point, there is a \(C^k\) extension on an open-neighborhood-of-the-point domain, which requires the codomain to be a subset of a \(C^\infty\) manifold without boundary. An issue because of which that proposition cannot be directly applied is that \(E\) may be with a nonempty boundary (when \(M\) is with a nonempty boundary), and another issue is that the extension is required to be a section, which that proposition does not guarantee. But the basic idea of this proposition is the same with that of that proposition: the rough reason for this proposition is that while \(E\) is locally \(U_p \times \mathbb{R}^k\), the boundary can exist only in the \(U_p\) part, but the extension needs to be the identity map with respect to the \(U_p\) part (because it is a section), and the concern is really about only the \(\mathbb{R}^k\) part, which has no boundary.
3: Proof
Whole Strategy: Step 1: take a chart trivializing subset of \(M\) around \(p\), with the chart, \((U_p \subseteq M, \phi_p)\), and the induced chart, \((\pi^{-1} (U_p) \subseteq E, \widetilde{\phi_p})\); Step 2: for the components function, \(f := \widetilde{\phi_p} \circ s \circ {\phi_p}^{-1} \vert_{\phi_p (U_p \cap S)}: \phi_p (U_p \cap S) \to \widetilde{\phi_p} (\pi^{-1} (U_p))\), take an open neighborhood of \(\phi_p (p)\), \(U_{\phi_p (p)} \subseteq \mathbb{R}^d\), and an extension of \(f\), \(f': U_{\phi_p (p)} \to \mathbb{R}^{d + k}\); Step 3: tweak \(f'\) to have \(f'': U_{\phi_p (p)} \to \mathbb{R}^k \times \phi_p (U_p)\); Step 4: take \(V_p := {\phi_p}^{-1} (\phi_p (U_p) \cap U_{\phi_p (p)})\) and \(s' := {\widetilde{\phi_p}}^{-1} \circ f'' \circ \phi_p \vert_{V_p} \to \pi^{-1} (V_p)\); Step 5: see that \(s'\) satisfies the requirements.
Step 1:
Let us take a chart trivializing subset of \(M\) around \(p\), with the chart, \((U_p \subseteq M, \phi_p)\), which is possible by the proposition that for any \(C^\infty\) vectors bundle, there is a chart trivializing open cover.
Let us take the induced chart, \((\pi^{-1} (U_p) \subseteq E, \widetilde{\phi_p})\), which is possible by the proposition that for any \(C^\infty\) vectors bundle, the trivialization of any chart trivializing open subset induces the canonical chart map.
\(s (U_p) \subseteq \pi^{-1} (U_p)\), because \(s\) is a section.
Step 2:
Let the components function be \(f := \widetilde{\phi_p} \circ s \circ {\phi_p}^{-1} \vert_{\phi_p (U_p \cap S)}: \phi_p (U_p \cap S) \to \widetilde{\phi_p} (\pi^{-1} (U_p))\).
There is an open neighborhood of \(\phi_p (p)\), \(U_{\phi_p (p)} \subseteq \mathbb{R}^d\), and a \(C^l\) extension of \(f\), \(f': U_{\phi_p (p)} \to \mathbb{R}^{d + k}\), because that is what the definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\) requires.
Step 3:
Denote \(f': (x^1, ..., x^d) \mapsto (f'^1 (x^1, ..., x^d), ..., f'^k (x^1, ..., x^d), f'^{k + 1} (x^1, ..., x^d), ..., , f'^{d + k} (x^1, ..., x^d))\). Each \(f'^j (x^1, ..., x^d)\) is \(C^l\).
Let us define \(f'': U_{\phi_p (p)} \to \mathbb{R}^k \times U_{\phi_p (p)}, (x^1, ..., x^d) \mapsto (f'^1 (x^1, ..., x^d), ..., f'^k (x^1, ..., x^d), x^1, ..., , x^d)\).
\(f''\) is obviously \(C^l\).
\(f \vert_{\phi_p (U_p \cap S)} = f' \vert_{\phi_p (U_p \cap S)} = f'' \vert_{\phi_p (U_p \cap S)}\), because \(f \vert_{\phi_p (U_p \cap S)} (x^1, ..., x^d) = f' \vert_{\phi_p (U_p \cap S)} (x^1, ..., x^d) = (f'^1 (x^1, ..., x^d), ..., f'^k (x^1, ..., x^d), f'^{k + 1} (x^1, ..., x^d), ..., , f'^{d + k} (x^1, ..., x^d)) = (f'^1 (x^1, ..., x^d), ..., f'^k (x^1, ..., x^d), x^1, ..., , x^d) = f'' \vert_{\phi_p (U_p \cap S)} (x^1, ..., x^d)\), because \(s\) is fiber-preserving and \(\widetilde{\phi_p}\) is induced from \(\phi_p\).
So, \(f''\) is a \(C^l\) extension of \(f\).
Step 4:
\(\phi_p (U_p) \cap U_{\phi_p (p)} \subseteq \phi_p (U_p)\) is an open neighborhood of \(\phi_p (p)\) on \(\phi_p (U_p)\).
Let us define \(V_p := {\phi_p}^{-1} (\phi_p (U_p) \cap U_{\phi_p (p)}) \subseteq M\) such that \(V_p \subseteq U_p\), which is an open neighborhood of \(p\).
Let us define \(s' := {\widetilde{\phi_p}}^{-1} \circ f'' \circ \phi_p \vert_{V_p}: V_p \to \pi^{-1} (V_p)\), which is possible because \(f'' \vert_{\phi_p \vert_{V_p} (V_p)}\) is into \(\mathbb{R}^k \times (U_{\phi_p (p)} \cap \phi_p (U_p)) \subseteq \mathbb{R}^k \times \phi_p (U_p) = \widetilde{\phi_p} (\pi^{-1} (U_p))\).
Step 5:
\(s'\) is indeed a section, obviously.
\(s'\) is \(C^l\), because while \(\phi_p \vert_{V_p}\) is \(C^\infty\) as \(: V_p \to \phi_p (V_p) \subseteq \mathbb{R}^d \text{ or } \mathbb{H}^d\), \(f''\) is \(C^l\) as \(: U_{\phi_p (p)} \subseteq \mathbb{R}^d \to \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k}\), and \(\widetilde{\phi_p}^{-1}\) is \(C^\infty\) as \(: \mathbb{R}^k \times \phi_p (U_p) \subseteq \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k} \to \pi^{-1} (U_p) \subseteq E\), the only concern for \(s'\) to be a legitimate chain of \(C^l\) maps is that \(\mathbb{R}^d \text{ or } \mathbb{H}^d\) for the codomain of \(\phi_p \vert_{V_p}\) is different from \(\mathbb{R}^d\) for the domain of \(f''\), but \(\phi_p \vert_{V_p}\) can be regarded to be \(\phi_p \vert_{V_p}: V_p \to \phi_p (V_p) \subseteq \mathbb{R}^d\), which obviously does not change \(C^\infty\)-ness, so, \(s'\) is indeed a legitimate chain of \(C^l\) maps, and is \(C^l\), 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.
\(s' \vert_{V_p \cap S} = s \vert_{V_p \cap S}\), because \(s' \vert_{V_p \cap S} = {\widetilde{\phi_p}}^{-1} \circ f'' \circ \phi_p \vert_{V_p} \vert_{V_p \cap S} = {\widetilde{\phi_p}}^{-1} \circ f \circ \phi_p \vert_{V_p} \vert_{V_p \cap S} = {\widetilde{\phi_p}}^{-1} \circ \widetilde{\phi_p} \circ s \circ {\phi_p}^{-1} \vert_{\phi_p (U_p \cap S)} \circ \phi_p \vert_{V_p} \vert_{V_p \cap S} = s \vert_{V_p} \vert_{V_p \cap S} = s \vert_{V_p \cap S}\).
References
<The previous article in this series | The table of contents of this series |
<The previous article in this series | The table of contents of this series | The next article in this series>
description/proof of that restricted \(c^\infty\) vectors bundle w.r.t. embedded submanifold with boundary is embedded submanifold with boundary
Topics
About:
\(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
-
The reader will have a description and a proof of the proposition that for any \(C^\infty\) vectors bundle, the restricted \(C^\infty\) vectors bundle w.r.t. any embedded submanifold with boundary is an embedded submanifold with boundary.
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:
\((E', M', \pi')\): \(\in \{\text{ the } C^\infty \text{ vectors bundles }\}\)
\(M\): \(\in \{\text{ the } d \text{ -dimensional embedded submanifolds with boundary of } M'\}\)
\((E, M, \pi)\): \(= \text{ the restricted } C^\infty \text{ vectors bundle }\)
//
Statements:
\(E \in \{\text{ the embedded submanifolds with boundary of } E'\}\)
//
2: Note
We already know that \(E\) is an immersed submanifold with boundary of \(E'\), by Note for the definition of restricted (C^\infty\) vectors bundle.
Definition-wise, \(E\) does not need to be an embedded submanifold with boundary of \(E'\) for an embedded submanifold with boundary \(M\), but in fact, \(E\) is an embedded submanifold with boundary of \(E'\), which we are going to prove.
3: Proof
Whole Strategy: Step 1: see that \(E\) has the subspace topology of \(E'\) by the proposition that for any topological space contained in any ambient topological space, if the space is ambient-space-wise locally topological subspace of the ambient space, the space is the topological subspace of the space; Step 2: see that the inclusion \(\iota: E \to E'\) is a \(C^\infty\) embedding.
Step 1:
Let us see that \(E\) has the subspace topology of \(E'\).
We are going to apply the proposition that for any topological space contained in any ambient topological space, if the space is ambient-space-wise locally topological subspace of the ambient space, the space is the topological subspace of the space.
Let \(m \in M\) be any.
There is a trivializing open subset around \(m\) for \(M'\), \(U'_m \subseteq M'\), such that \(U_m := U'_m \cap M\) is a chart domain for \(M\), by the proposition that for any \(C^\infty\) manifold with boundary and its any embedded submanifold with boundary, around each point on the submanifold with boundary, there is a trivializing open subset for the manifold with boundary whose intersection with the submanifold with boundary is a chart domain for the submanifold with boundary. Let the chart be \((U_m \subseteq M, \phi_m)\).
The corresponding \(U'_\beta\) s and \((U_\beta \subseteq M, \phi_\beta)\) s are legitimate as the ones used for constructing the topology and the atlas of \(E\) in the definition of restricted \(C^\infty\) vectors bundle: \(U_\beta\) is an embedded submanifold with boundary of \(M\), by Note for the definition of open submanifold with boundary of \(C^\infty\) manifold with boundary and \(U_\beta\) is an embedded submanifold with boundary of \(M'\), by 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. We take \(\{(U_\beta \subseteq M, \phi_\beta) \vert \beta \in B\}\) and \(\{U'_\beta \vert \beta \in B\}\) and \(\{\Phi'_\beta: \pi'^{-1} (U'_\beta) \to U'_\beta \times \mathbb{R}^k \vert \beta \in B\}\), accordingly.
For each point on \(E\), let us take the open neighborhood of the point on \(E'\) for the proposition that for any topological space contained in any ambient topological space, if the space is ambient-space-wise locally topological subspace of the ambient space, the space is the topological subspace of the space as \(\pi'^{-1} (U'_\beta)\).
Let us see that \(\pi'^{-1} (U'_\beta)\) is indeed one the theorem requires.
\(\pi'^{-1} (U'_\beta)\) is indeed an open neighborhood of the point on \(E'\), because the point is contained in it and \(U'_\beta\) is open on \(M'\) and \(\pi'\) is continuous.
Is \(\pi'^{-1} (U'_\beta) \cap E\) an open subset of \(E\)?
Yes, because \(\pi'^{-1} (U'_\beta) \cap E = \pi'^{-1} (U'_\beta) \cap \pi'^{-1} (M) = \pi'^{-1} (U'_\beta \cap M)\), by the proposition that for any map, the map preimage of any intersection of sets is the intersection of the map preimages of the sets, \( = \pi^{-1} (U_\beta)\), which is open on \(E\), because \(U_\beta\) is open on \(M\) and \(\pi\) is continuous.
Does \(\pi'^{-1} (U'_\beta) \cap E \subseteq E \text{ as the subspace of } E\) equal \(\pi'^{-1} (U'_\beta) \cap E \subseteq \pi'^{-1} (U'_\beta) \text{ as the subspace of } \pi'^{-1} (U'_\beta)\)?
\(\pi'^{-1} (U'_\beta) \cap E \subseteq E\) is nothing but \(\pi^{-1} (U_\beta)\), and \(\Phi_\beta: \pi^{-1} (U_\beta) \to U_\beta \times \mathbb{R}^k\) is homeomorphic.
For each subset, \(S \subseteq \pi'^{-1} (U'_\beta) \cap E\), \(S \cap E = S \cap \pi^{-1} (U_\beta)\), because while \(S \cap \pi^{-1} (U_\beta) \subseteq S \cap E\) is obvious, for each \(p \in S \cap E\), \(p \in S \subseteq \pi'^{-1} (U'_\beta)\), which implies that \(\pi' (p) = \pi (p) \in U'_\beta\), but also \(\pi (p) \in M\), so, \(\pi (p) \in U'_\beta \cap M = U_\beta\), so, \(p \in \pi^{-1} (U_\beta)\), and \(p \in S \cap \pi^{-1} (U_\beta)\), and so, \(S \cap E \subseteq S \cap \pi^{-1} (U_\beta)\).
Especially, \(\pi'^{-1} (U'_\beta) \cap E = \pi'^{-1} (U'_\beta) \cap \pi^{-1} (U_\beta)\).
\(\Phi'_\beta: \pi'^{-1} (U'_\beta) \to U'_\beta \times \mathbb{R}^k\) is homeomorphic, and \(\Phi'_\beta \vert_{\pi'^{-1} (U'_\beta) \cap E}: \pi'^{-1} (U'_\beta) \cap E = \pi'^{-1} (U'_\beta) \cap \pi^{-1} (U_\beta) \to \Phi'_\beta (\pi'^{-1} (U'_\beta) \cap \pi^{-1} (U_\beta)) = \Phi'_\beta (\pi^{-1} (U_\beta)) = \Phi_\beta (\pi^{-1} (U_\beta)) = U_\beta \times \mathbb{R}^k \subseteq U'_\beta \times \mathbb{R}^k\) is homeomorphic, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous. Note that it is homeomorphic with the domain, \(\pi'^{-1} (U'_\beta) \cap E\), regarded as the subspace of \(\pi'^{-1} (U'_\beta)\) and the codomain, \(U_\beta \times \mathbb{R}^k\), regarded as the subspace of \(U'_\beta \times \mathbb{R}^k\).
Let us think of the identity map, \(id: U_\beta \times \mathbb{R}^k \to U_\beta \times \mathbb{R}^k \subseteq U'_\beta \times \mathbb{R}^k\). In fact, \(U_\beta \times \mathbb{R}^k \subseteq U'_\beta \times \mathbb{R}^k\) regarded as the subspace of \(U'_\beta \times \mathbb{R}^k\) is nothing but \(U_\beta \times \mathbb{R}^k\), because \(M\) is a topological subspace of \(M'\).
Now, for each open subset of \(\pi'^{-1} (U'_\beta) \cap E \subseteq E = \pi^{-1} (U_\beta)\), \(U\), \(\Phi_\beta (U) \subseteq U_\beta \times \mathbb{R}^k\) is open; \(id (\Phi_\beta (U)) \subseteq U_\beta \times \mathbb{R}^k \subseteq U'_\beta \times \mathbb{R}^k\) is open; \({\Phi'_\beta \vert_{\pi'^{-1} (U'_\beta) \cap E}}^{-1} (id (\Phi_\beta (U))) \subseteq \pi'^{-1} (U'_\beta) \cap E \subseteq \pi'^{-1} (U'_\beta)\) is open. In fact, that \({\Phi'_\beta \vert_{\pi'^{-1} (U'_\beta) \cap E}}^{-1} \circ id \circ \Phi_\beta\) is the identity map, because \(\Phi_\beta\) is the restriction of \(\Phi'_\beta\). That shows that each open subset of \(\pi'^{-1} (U'_\beta) \cap E \subseteq E\) is open on \(\pi'^{-1} (U'_\beta) \cap E \subseteq \pi'^{-1} (U'_\beta)\).
Likewise, for each open subset of \(\pi'^{-1} (U'_\beta) \cap E \subseteq \pi'^{-1} (U'_\beta)\), \(U\), \(\Phi'_\beta \vert_{\pi'^{-1} (U'_\beta) \cap E} (U) \subseteq U_\beta \times \mathbb{R}^k \subseteq U'_\beta \times \mathbb{R}^k\) is open; \({id}^{-1} (\Phi'_\beta \vert_{\pi'^{-1} (U'_\beta) \cap E} (U)) \subseteq U_\beta \times \mathbb{R}^k\) is open; \({\Phi_\beta}^{-1} ({id}^{-1} (\Phi'_\beta \vert_{\pi'^{-1} (U'_\beta) \cap E} (U)) \subseteq \pi'^{-1} (U'_\beta) \cap E \subseteq E\) is open. In fact, that \({\Phi_\beta}^{-1} \circ {id}^{-1} \circ \Phi'_\beta \vert_{\pi'^{-1} (U'_\beta) \cap E}\) is the identity map, because \(\Phi_\beta\) is the restriction of \(\Phi'_\beta\). That shows that each open subset of \(\pi'^{-1} (U'_\beta) \cap E \subseteq \pi'^{-1} (U'_\beta)\) is open on \(\pi'^{-1} (U'_\beta) \cap E \subseteq E\).
So, yes, \(\pi'^{-1} (U'_\beta) \cap E \subseteq E \text{ as the subspace of } E\) equals \(\pi'^{-1} (U'_\beta) \cap E \subseteq \pi'^{-1} (U'_\beta) \text{ as the subspace of } \pi'^{-1} (U'_\beta)\).
So, \(E\) is the topological subspace of \(E'\).
Step 2:
Let us see that the inclusion \(\iota: E \to E'\) is a \(C^\infty\) embedding.
We already know that \(\iota\) is a \(C^\infty\) immersion, because \(E\) is an immersed submanifold with boundary of \(E'\).
The codomain restriction, \(\iota': E \to \iota (E) \subseteq E'\), is homeomorphic, because \(E\) has the subspace topology of \(E'\) by Step 1, and \(\iota'\) is the identity map.
So, \(\iota\) is a \(C^\infty\) embedding.
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
definition of Sylow p-subgroup of group
Topics
About:
group
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of Sylow p-subgroup of group.
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:
\( G\): \(\in \{\text{ the groups }\}\)
\( p\): \(\in \{\text{ the prime numbers }\}\)
\( B\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(*G_{p, \beta}\): \(\in \{\text{ the maximal p-subgroups of } G\}\), \(\beta \in B\)
//
Conditions:
//
"maximal p-subgroup" means that it is a p-subgroup and there is no p-subgroup that contains it.
\(Syl_p (G) := \{G_{p, \beta} \vert \beta \in B\}\), which is the set of all the Sylow p-subgroups of \(G\).
2: Note
For an arbitrary \(p\), there may be no Sylow p-subgroup, because there may be no p-subgroup.
If there is a p-subgroup, \(H\), there will be a Sylow p-subgroup that contains \(H\), which is by Zorn's lemma: let \(A\) be the set of the p-subgroups that contains \(H\); let \(B\) be any nonempty chain of \(A\); \(\cup B \in A\), because for each \(g_1, g_2 \in \cup B\), \(g_1 \in C \in B\) and \(g_2 \in D \in B\), but \(C \subseteq D\) or \(D \subseteq C\); supposing without loss of generality that \(C \subseteq D\), \(g_1, g_2 \in D\); \(g_1 g_2 \in D \in \cup B\); \(1 \in C \in \cup B\); for each \(g \in \cup B\), \(g \in C\), \(g^{-1} \in C\); so, \(\cup B\) is a subgroup; for each \(g \in \cup B\), \(g \in C\), and \(g\) has order of a power of \(p\); \(H \subseteq \cup B\); so, \(\cup B\) is a p-subgroup that contains \(H\), which means that \(\cup B \in A\); then, Zorn's lemma says that there is a maximal p-subgroup that contains \(H\).
When \(G\) is finite, \(\vert G \vert = p_1^{n_1} ... p_k^{n_k}\) for some prime numbers, \(p_1 \lt ... \lt p_k\), and some \(n_1, ..., n_k \in \mathbb{N} \setminus \{0\}\). The order of each Sylow \(p_j\)-subgroup of \(G\) is \(p_j^{n_j}\), which is a part of the Sylow theorem.
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
definition of normalizer of subgroup on group
Topics
About:
group
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of normalizer of subgroup on group.
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:
\( G'\): \(\in \{\text{ the groups }\}\)
\( G\): \(\in \{\text{ the subgroups of } G'\}\)
\(*N_{G'} (G)\): \(= \{g' \in G' \vert g' G g'^{-1} = G\}\)
//
Conditions:
//
2: Note
The name has originated from the fact that \(N_{G'} (G)\) is the largest subgroup of \(G'\) of which (\(N_{G'} (G)\)) \(G\) is a normal subgroup.
Let us confirm the fact.
Let us see that \(N_{G'} (G)\) is a group.
For each \(h_1, h_2 \in N_{G'} (G)\), \(h_1 h_2 \in N_{G'} (G)\), because \(h_1 h_2 G (h_1 h_2)^{-1} = h_1 h_2 G {h_2}^{-1} {h_1}^{-1} = h_1 G {h_1}^{-1} = G\).
\(1 \in N_{G'} (G)\), because \(1 G 1^{-1} = G\).
For each \(h \in N_{G'} (G)\), \(h^{-1} \in N_{G'} (G)\), because \(h^{-1} G h = h^{-1} (h G h^{-1}) h = (h^{-1} h) G (h^{-1} h) = 1 G 1 = G\).
Associativity holds because it holds in the ambient \(G'\).
\(G \subseteq N_{G'} (G)\), because for each \(g \in G\), \(g G g^{-1} = G\).
\(G\) is a normal subgroup of \(N_{G'} (G)\), because for each \(h \in N_{G'} (G)\), \(h G h^{-1}\).
\(N_{G'} (G)\) is such the largest, because if there is a subgroup of \(G'\), \(G''\), such that \(G\) is a normal subgroup of \(G''\), for each \(g'' \in G''\), \(g'' G g''^{-1} = G\), which means that \(g'' \in N_{G'} (G)\), which means that \(G'' \subseteq N_{G'} (G)\).
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
definition of field
Topics
About:
field
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of field
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:
\(*F\): \(\in \{\text{ the rings }\}\)
//
Conditions:
\(\forall r_1, r_2 \in F (r_1 r_2 = r_2 r_1)\)
\(\land\)
\(\forall r \in F (\exists r' \in F (r r' = r' r = 1))\)
//
2: Note
Inevitably, such \(r'\) is unique for fixed \(r\), because supposing there is another \(r'' \in F\) such that \(r r'' = r'' r = 1\), from \(r' r = 1\), \(r' r r'' = 1 r'' = r''\), but \(r' r r'' = r' 1 = r'\), and so, \(r' = r''\).
As \(r'\) is unique for \(r\), it is warranted to be denoted as \(r^{-1}\), called "the inverse of \(r\)".
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series |
Japanese schools talk about "shugo"s, but are they sets or collections?
Topics
About:
elementary school mathematics
The table of contents of this article
Starting Context
Target Context
-
The reader will know the distinction between sets and collections.
Orientation
There is an article on becoming a benefactor of humanity by being a conduit of truths
There is an article on definition of set and Russel's "paradox".
Main Body
1: What Does "shugo" (Used in Japanese Schools) Mean?
Special-Student-7-Hypothesizer
Japanese schools talk about "shugo" s.
For example, they think of the "shugo" of some 3 apples, the "shugo" of the students of a class, etc..
Special-Student-7-Rebutter
What is "shugo" exactly?
Special-Student-7-Hypothesizer
Any "shugo" is any collection of some objects such that the membership is unambiguous.
Special-Student-7-Rebutter
Is there ever a collection such that the membership is ambiguous?
Special-Student-7-Hypothesizer
I do not think there is: the concept of 'collection' itself includes the requirement that the membership is unambiguous; the qualification is just for emphasis.
Special-Student-7-Rebutter
So, is "shugo" nothing but 'collection' in English?
Special-Student-7-Hypothesizer
I can just guess so, because Japanese textbooks do not say what "shugo" is called in English, but dictionaries usually say that "shugo" in mathematics is "set" in English.
Special-Student-7-Rebutter
But 'set' and 'collection' are 2 different things in mathematics.
2: Caution: There Are Some Multiple Set Theories
Special-Student-7-Hypothesizer
As a caution, there are some multiple set theories, and what 'set' is in each theory depends on the theory.
Special-Student-7-Rebutter
Very confusing; cannot the theories use different names?
Special-Student-7-Hypothesizer
It is very confusing, but as each theory exists because it regards the other theories unsatisfactory, each theory says "'set' we call is the real 'set'!", and each theory uses "set".
Special-Student-7-Rebutter
I kind of understand, but ...
Special-Student-7-Hypothesizer
Anyway, arguably the most popular one is the ZFC set theory, and we will mean the ZFC set theory by "the set theory" and mean 'set' in the ZFC set theory by "set" hereafter.
Note that what we are going to say here are not particularly canonical; as we have not seen any convincing argument, we are presenting a hypothesis that is convincing for us. Nobody should accept something just because someone (whoever he or she is) says so or just because many people say so; check yourself whether the hypothesis is unboundedly consistent, which is the only way to approach truths.
3: How Have 'Set' and 'Collection' Ended Up Meaning Different Things?
Special-Student-7-Hypothesizer
Most textbooks cite Russel's "paradox" as the reason why the set theory needs to be as it is, but the arguments are not convincing at all, at least for us.
Special-Student-7-Rebutter
What are the arguments like?
Special-Student-7-Hypothesizer
Historically, there was the so-called "naive set theory", which is based on the so-called "naive comprehension axiom", which is "Any precisely specified property of member can be used to define a set.".
Then, Russel's "paradox" came along and refuted the "naive comprehension axiom", and so refuted the "naive set theory".
So, now, the ZFC set theory admits only the empty set and the things constructed from the empty set as "set".
So, the collection of some 2 electrons are not admitted to be "set", because the collection is not constructed from the empty set.
Special-Student-7-Rebutter
There seems a wide gap in the reasoning: why would the collection of the 2 electrons not be "set" just because the "naive comprehension axiom" has been refuted?
Special-Student-7-Hypothesizer
I have not seen any convincing explanation.
In fact, why will we not have the modified axiom that "Any precisely specified property of member that (the property) determines membership unambiguously can be used to define a set."?
Special-Student-7-Rebutter
What Russel's "paradox" says is that just "precisely specified property of member" does not guarantee the unambiguous-ness of membership, so, why will we not add the unambiguous-ness as an additional requirement?
Special-Student-7-Hypothesizer
I have not seen any convincing argument why; it seems just that as checking it is not easy in general, they want a way to be able to define sets without bothering to do checking.
Special-Student-7-Rebutter
Well, is that an attitude to humbly approach truths? That seems an attitude to ignore whatever are inconvenient for humans.
Special-Student-7-Hypothesizer
Whether the unambiguous-ness can be checked easily is just a matter of human convenience, and the essence of the ZFC set theory is that it gave up covering the concept of "collection" and has decided to degenerate to deal with only convenient-for-humans kind of collections, which are now called "sets", which is my understanding.
In fact, I do not say at all that having a theory about such limited kind of collections is bad, but we need to be aware what the theory is really doing.
Special-Student-7-Rebutter
We need to avoid some confusions: "naive set theory" means the theory with the "naive comprehension axiom" not the concept of 'collection'; while the ZFC set theory excludes most collections like a collection of some 2 electrons, that is not because the collection of 2 electrons are inappropriate but because the set theory has degenerated.
Special-Student-7-Hypothesizer
The main cause of confusions is that the ZFC set theory keeps using the term, "set", while the concept of 'set' has degenerated from equaling 'collection' to 'convenient-for-humans collection'.
4: In Fact, a Collection in General Is Not Defined by Any Property of Member
Special-Student-7-Hypothesizer
In fact, I think that the "naive comprehension axiom" is more fundamentally problematic than Russel's "paradox".
The problem is that it tries to define any collection by a property of member.
For example, let us think of the collection of some balls I put into a bag.
I put a ball into the bag just because my groping hand happened to touch the ball, not because the ball is red or blue or something.
Special-Student-7-Rebutter
You mean that the collection is not defined by any property of ball, like being red or something?
Special-Student-7-Hypothesizer
Yes.
Special-Student-7-Rebutter
Your collection is usually defined by enumerating the balls, as the collection is finite: you cannot put infinitely many balls into the bag by groping.
Special-Student-7-Hypothesizer
Yes, when the collection is finite, there is the escape.
But what if the collection is infinite?
Special-Student-7-Rebutter
How will you define an infinite collection not by property of member?
Special-Student-7-Hypothesizer
If we ignore Relativity, let us take a \(\mathbb{R}^3\) Cartesian coordinates system for the Universe, and take the 1-radius open ball around each rational-coordinates point. Then, the numbers of the electrons in each open ball at a time, which is a subcollection of the natural numbers set, is my infinite collection.
Special-Student-7-Rebutter
You mean that the collection is not defined by any property of natural number, like being even or something?
Special-Student-7-Hypothesizer
Yes.
Special-Student-7-Rebutter
It does not seem to be really infinite, supposing that there are only some finite number of electrons in the Universe.
Special-Student-7-Hypothesizer
Then, let us take the pair of the center-coordinates and the number of the electrons, like ((1.2, 3.4, 5.6), 7).
Special-Student-7-Rebutter
Then, certainly, the collection is infinite.
What if we are not allowed to ignore Relativity?
Special-Student-7-Hypothesizer
Let us take a chart for the spacetime manifold, whose (the chart's) domain is homeomorphic to \(\mathbb{R}^4\), and let us do the same thing for that \(\mathbb{R}^4\).
Special-Student-7-Rebutter
So, you are saying that such collections exist in the reality but the "naive comprehension axiom" cannot grasp such collections.
Special-Student-7-Hypothesizer
In fact, that is also the problem of the "restricted comprehension axiom", which is in fact the same with the "naive comprehension axiom" except that the "restricted comprehension axiom" thinks of only the elements of an already-known-to-be-set collection.
Special-Student-7-Rebutter
Is "the number of the electrons in the open ball" not 'property of member'?
Special-Student-7-Hypothesizer
At least, the ZFC set theory does not admit such a property: the "restricted comprehension axiom" requires that the property is expressed as a formula that allows only the specified operators like \(\in\) and some already-known-to-be-set collections.
Special-Student-7-Rebutter
Did the "naive comprehension axiom" admit such a property?
Special-Student-7-Hypothesizer
Maybe, but such a usage of the term, "property of member", would be quite harmful (a too-far-fetched interpretation, I would say), in my opinion. Being "the number of the electrons in the open ball" is not any property of the natural number but is a property of the Universe: I mean, being "the number of the electrons in the open ball" is not about the natures of the natural number (like being even, prime, or something) but about the natures of the Universe.
Special-Student-7-Rebutter
Anyway, the "restricted comprehension axiom" does not allow your collection.
Special-Student-7-Hypothesizer
So, there is the collection in the reality, but the ZFC set theory refuses to cope with the collection.
5: "shugo" Means 'Collection'
Special-Student-7-Hypothesizer
So, "shugo" in Japanese textbooks means 'collection' not "set".
And when a Japanese student later hears of Russel's "paradox" and hears '"set" of some 2 electrons' refuted, the "shugo" of the 2 electrons is not refuted at all, because the collection of the 2 electrons is not refuted at all.
And when he or she later hears that "naive set theory" is inappropriate, the concept of "shugo" (the concept of 'collection') is not inappropriate at all, because the concept of 'collection' is not based on the naive comprehension axiom.
6: The Concept of 'Collection' Lives on
Special-Student-7-Hypothesizer
The concept of 'collection' is still valid and is indeed used in mathematics.
As a typical example, in the category theory, a category in general is not any set but a collection. For example, the category of all the sets, \(Set\), is a collection but not any set.
Special-Student-7-Rebutter
\(Set\) is prevalently called "class".
Special-Student-7-Hypothesizer
Yes, 'class' is a concept wider than 'set' and narrower than 'collection', but anyway, a class is not any set in general.
Special-Student-7-Rebutter
How is 'class' narrower than 'collection'?
Special-Student-7-Hypothesizer
Roughly speaking, any class is still constructed from the empty set and is still defined by property of member with a formula.
Anyway, if something was inappropriate just because it was not "set" in the ZFC set theory, the whole category theory would be inappropriate.
And "collection" is still frequently used in many mathematical textbooks: while a reader may wonder why "collection" is being used instead of "set", that is because the mentioned object is not or is not necessarily any "set" according to the ZFC set theory, and if 'collection' was inappropriate, such textbooks would be inappropriate, which is not the case.
7: Mathematics Is Not Exactly Built on the Set Theory
Special-Student-7-Hypothesizer
A rather prevalent misconception is that "whole mathematics is built on the set theory.".
Special-Student-7-Rebutter
That is what the naive set theory aspired to do.
Special-Student-7-Hypothesizer
But the attempt failed (by mainly Russel's "paradox") and because the set theory gave up dealing with general collections and has degenerated to be a theory on a very limited kind of collections, the aspiration is given up.
Certainly, the set theory is still a very important part of mathematics, but it is not exactly the basis of whole mathematics.
Special-Student-7-Rebutter
A proof is that "collection" is still used in many mathematical textbooks.
References
<The previous article in this series | The table of contents of this series |
<The previous article in this series | The table of contents of this series | The next article in this series>
definition of integers modulo natural number group
Topics
About:
group
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of integers modulo natural number group.
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:
\( \mathbb{Z}\): \(= \text{ the integers set }\)
\( n\): \(\in \mathbb{N}\)
\( \sim\): \(\in \{\text{ the equivalence relations on } \mathbb{Z}\}\), such that \(\forall z_1, z_2 \in \mathbb{Z} (\exists l \in \mathbb{Z} (z_1 = z_2 + l n) \iff z_1 \sim z_2)\)
\(*\mathbb{Z} / n\): \(= \mathbb{Z} / \sim\) as the quotient set with the group operation specified below
//
Conditions:
\(\forall [z_1], [z_2] \in \mathbb{Z} / n ([z_1] + [z_2] = [z_1 + z_2])\)
//
2: Note
The operation is usually denoted as \(+\) because it is based on \(+\) on \(\mathbb{Z}\).
Let us see that the operation is well-defined.
That is about that \([z_1 + z_2]\) does not depend on the representatives, \(z_1, z_2\).
Let \(z'_1, z'_2 \in \mathbb{Z}\) be such that \([z_1] = [z'_1]\) and \([z_2] = [z'_2]\). That means that \(z'_1 = z_1 + l_1 n\) and \(z'_2 = z_2 + l_2 n\). \([z'_1 + z'_2] = [z_1 + l_1 n + z_2 + l_2 n] = [z_1 + z_2 + (l_1 + l_2) n] = [z_1 + z_2]\).
Let us see that \(\mathbb{Z} / n\) is indeed a Abelian group.
The operation is closed, because \([z_1 + z_2] \in \mathbb{Z} / n\).
\([0]\) is the identity element: \([0] + [z] = [0 + z] = [z]\) and \([z] + [0] = [z + 0] = [z]\).
For each \([z]\), \([- z]\) is the inverse: \([z] + [- z] = [z - z] = [0]\) and \([- z] + [z] = [- z + z] = [0]\).
The operation is associative: for each \([z_1], [z_2], [z_3] \in \mathbb{Z} / n\), \(([z_1] + [z_2]) + [z_3] = [z_1 + z_2] + [z_3] = [z_1 + z_2 + z_3] = [z_1] + [z_2 + z_3] = [z_1] + ([z_2] + [z_3])\).
So, \(\mathbb{Z} / n\) is a group.
\(\mathbb{Z} / n\) is Abelian: for each \([z_1], [z_2] \in \mathbb{Z} / n\), \([z_1] + [z_2] = [z_1 + z_2] = [z_2 + z_1] = [z_2] + [z_1]\).
Obviously, \(\mathbb{Z} / n = \{[0], ..., [n - 1]\}\).
References
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<The previous article in this series | The table of contents of this series | The next article in this series>
definition of left or right coset of subgroup by element of group
Topics
About:
group
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of left or right coset of subgroup by element of group.
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:
\( G'\): \(\in \{\text{ the groups }\}\)
\( G\): \(\in \{\text{ the subgroups of } G'\}\)
\( g'\): \(\in G'\)
\(*g' G\): \(= \text{ the left coset of } G \text{ by } g'\)
\(*G g'\): \(= \text{ the right coset of } G \text{ by } g'\)
//
Conditions:
//
2: Note
For any \(g'_1, g'_2 \in G'\), \(g'_1 G \cap g'_2 G = \emptyset\) or \(g'_1 G = g'_2 G\): when \(g'_1 G \cap g'_2 G \neq \emptyset\), there is a \(g'_3 \in g'_1 G \cap g'_2 G\), and \(g'_1 G = g'_3 G = g'_2 G\), by the proposition that with respect to any subgroup, the coset by any element of the group equals a coset if and only if the element is a member of the latter coset, whether they are left cosets or right cosets.
Likewise, for any \(g'_1, g'_2 \in G'\), \(G g'_1 \cap G g'_2 = \emptyset\) or \(G g'_1 = G g'_2\).
\(\vert g' G \vert = \vert G \vert\): for each \(g_1, g_2 \in G\) such that \(g_1 \neq g_2\), \(g' g_1 \neq g' g_2\), because supposing that \(g' g_1 = g' g_2\), \(g_1 = {g'}^{-1} g' g_1 = {g'}^{-1} g' g_2 = g_2\), a contradiction.
Likewise, \(\vert G g' \vert = \vert G \vert\).
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
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.
References
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