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Theorem 21.2. (Erdös-Rado) Let \( \alpha ,\gamma \) be cardinals with \( \gamma \geq {\aleph }_{0} \) . Then \( \left( {\forall x}\right) \left\lbrack {x \in A \rightarrow \overline{\bar{x}} \leq \alpha }\right\rbrack \land \left( {\forall x \subseteq A}\right) \left\lbrack {x\text{ is quasi-disjoint } \rightarrow \ove... | Proof. For each \( \delta < {\alpha }^{ + } \) we construct a set \( {A}_{\delta } \subseteq A \) such that\n\na. \( \left( {\forall \delta < {\alpha }^{ + }}\right) \left\lbrack {{\overline{\bar{A}}}_{\delta } \leq {\gamma }^{\alpha }}\right\rbrack \) and\n\nb. \( A = \mathop{\bigcup }\limits_{{\delta < {\alpha }^{ + ... | Yes |
Theorem 21.3. Let \( \alpha ,\gamma \) be cardinals with \( \gamma \geq {\aleph }_{0} \) . Suppose that \( A = \) \( \left\{ {{A}_{t} \mid t \in T}\right\} \) and \( \left\{ {{B}_{t} \mid t \in T}\right\} \) satisfy the following conditions:\n\n1. \( \left( {\forall t \in T}\right) \left\lbrack {{\overline{\bar{A}}}_{t... | Proof. Note that \( \overline{\bar{A}} = \overline{\bar{T}} \) by 2 . We will show that \( A \) satisfies the conditions of Theorem 21.2. Let \( \left\{ {{A}_{t} \mid t \in {T}_{0}}\right\} \) be quasi-disjoint, \( 0 \neq {T}_{0} \subseteq T \) . Choose \( {t}_{0} \in {T}_{0} \) and define\n\n\[ \n{C}_{t} = {A}_{t} \ca... | Yes |
Corollary 21.4. Let \( \gamma \) be a cardinal with \( \gamma \geq {\aleph }_{0} \) . If\n\n1. \( \left( {\forall t \in T}\right) \left\lbrack {{\overline{\bar{A}}}_{t} < \omega \land {\overline{\bar{B}}}_{t} \leq \gamma }\right\rbrack \) .\n\n2. \( \left( {\forall t,{t}^{\prime } \in T}\right) \left\lbrack {t \neq {t}... | Proof. Let \( {A}^{\left( n\right) } = \left\{ {{A}_{t} \mid t \in T \land {\bar{A}}_{t} = n}\right\} \) for \( n \in \omega \) . By Theorem 21.3 (for \( \alpha = n \) ), \( {\bar{A}}^{\left( n\right) } = {\gamma }^{n} = \gamma \) .\n\nSince \( A = \left\{ {{A}_{t} \mid t \in T}\right\} = \mathop{\bigcup }\limits_{{n <... | Yes |
Theorem 21.5. (Marczewski) Let \( I \) be a set and \( \left\{ {{X}_{i} \mid i \in I}\right\} \) be a family of topological spaces such that each \( {X}_{i} \) has a base \( {b}_{i} \) of cardinality \( \leq \gamma \) . Let \( X = \mathop{\prod }\limits_{{i \in I}}{X}_{i} \) be the product space. If \( \left\{ {{O}^{\l... | Proof. We can assume that \( {X}_{i} \cap {X}_{{i}^{\prime }} = 0 \) for \( i \neq {i}^{\prime } \) and \( {X}_{i} \in {b}_{i} \) for \( i,{i}^{\prime } \in I \) . Let\n\n\[ \n{p}_{j} : \mathop{\prod }\limits_{{i \in I}}{X}_{i} \rightarrow {X}_{j},\;j \in J, \n\] \n\nbe the canonical projection, \n\n\[ \n{O}_{j}^{\left... | Yes |
1. \( A \subseteq S \rightarrow {A}^{-S} = {A}^{ - } \cap S \) . | Proof. 1. Let \( A \subseteq S \) ; then obviously\n\n\[ \n{A}^{-s} \subseteq {A}^{ - } \cap S \n\]\n\nConversely, let \( p \in {A}^{ - } \cap S \) . Then\n\n\[ \np \in S \land \left( {\forall N\left( p\right) }\right) \left\lbrack {N\left( p\right) \cap A \neq 0}\right\rbrack \n\]\n\n\[ \n\left( {\forall N\left( p\rig... | Yes |
Theorem 22.2. If \( A \subseteq S \), if \( S \) is dense in \( X \), and if \( A \) is regular open in \( S \) then \( A = {A}^{-0} \cap S \) . | Proof. Let \( A \subseteq S \) be regular open in \( S \) . By Theorem 22.1.1, \( {A}^{-S} = \) \( {A}^{ - } \cap S \) . Then, since \( {A}^{-0} \cap S \) is open in \( S \) ,\n\n\[ \n{A}^{-0} \cap S \subseteq {\left( {A}^{-S}\right) }^{0S} = A.\n\]\n\nOn the other hand, if \( p \in A \), then since \( A \) is open in ... | Yes |
Theorem 22.3. Let \( S \) be dense in \( X \) and let \( A, B \) be regular open (in \( X \) ). Then \( A \cap S \subseteq B \cap S \rightarrow A \subseteq B \) . | Proof. \( A = A \cap {S}^{ - } \subseteq {\left( A \cap S\right) }^{ - } \subseteq {\left( B \cap S\right) }^{ - } \subseteq {B}^{ - } \n\n\[ \nA \subseteq {B}^{-0}\;\text{since}A\text{is open} \n\] \n\n\[ \nA \subseteq B\;\text{ since }{B}^{-0} = B. \n\] | Yes |
Theorem 22.4. If \( S \) is dense in \( X \) and \( A \) is regular open (in \( X \) ) then \( A \cap S \) is regular open in \( S \) . | Proof. We need only show that \( {\left( A \cap S\right) }^{-{S0S}} \subseteq {A}^{-0} \cap S \) for by the proof of Theorem 22.2 we know that the reverse inclusion holds. Let\n\n\[ p \in {\left( A \cap S\right) }^{-{S0S}}, \]\n\nthen\n\n\[ \left( {\exists N\left( p\right) }\right) \left\lbrack {N\left( p\right) \cap S... | Yes |
Theorem 22.5. Let \( S \) be dense in \( X \) and let \( \mathbf{B} \) and \( {\mathbf{B}}_{0} \) be the complete Boolean algebras of all regular open sets in \( X \) and \( S \) respectively. Then \( {\mathbf{B}}_{0} \) and \( \mathbf{B} \) are isomorphic. An isomorphism \( i : {\mathbf{B}}_{0} \rightarrow \mathbf{B} ... | Proof. For \( {b}_{0} \in {B}_{0} \), define \( i\left( {b}_{0}\right) = {b}_{0}{}^{-0} \) . Then \( i : {B}_{0} \rightarrow B \) and \( {b}_{0} = \) \( i\left( {b}_{0}\right) \cap S \), by Theorem 22.2. Let \( b \in B \) . Since \( b \cap S \) is regular open in \( S \) (by Theorem 22.4), \( i\left( {b \cap S}\right) ... | Yes |
Theorem 22.7. Let \( \mathbf{B} \) be a Boolean algebra (not necessarily complete) and let \( \mathbf{P} = \langle P, \leq \rangle \) be the partial order structure determined by \( \mathbf{B} \), i.e., \( P = B - \{ \mathbf{0}\} \) and \( \leq \) is \( \leq \) in \( \mathbf{B} \). Let \( \widetilde{\mathbf{B}} \) be t... | Proof. \( {j}^{cc}P \) is dense in \( \widetilde{\mathbf{B}} \) because \( \{ \left\lbrack p\right\rbrack \mid p \in P\} \) is a base for open sets in P. Repeating the proof of Theorem 1.30 we see that \( \langle \widetilde{\mathbf{B}}, j\rangle \) is a completion of B. If \( \left\langle {{\widetilde{\mathbf{B}}}_{1},... | Yes |
Theorem 22.8. Let \( X, Y \) be topological spaces and let \( f : X \rightarrow Y \) be an open continuous map onto \( Y \) . Then for \( B \subseteq Y \) we have,\n\n1. \( {\left. {\left( {f}^{-1}\right) }^{cc}\left( {B}^{ - }\right) = \left( {\left( {f}^{-1}\right) }^{cc}B\right) \right) }^{ - } \) .\n\n2. \( {\left(... | Proof. Let \( x \in {\left( {f}^{-1}\right) }^{ii}\left( {B}^{ - }\right) \) . Then \( f\left( x\right) \in {B}^{ - } \) and hence\n\n\[ \left( {\forall N\left( {f\left( x\right) }\right) }\right) \left\lbrack {N\left( {f\left( x\right) }\right) \cap B \neq 0}\right\rbrack \]\n\n\[ \left( {\forall N\left( x\right) }\ri... | Yes |
Theorem 22.10.\n\n1. \( {b}_{1} \leq i\left( {\# \left( {b}_{1}\right) }\right) \) .\n\n2. \( {b}_{1} = \mathbf{0} \rightarrow \# \left( {b}_{1}\right) = \mathbf{0} \) .\n\n3. \( \# \left( {i\left( {b}_{0}\right) }\right) = {b}_{0} \) .\n\n4. \( {b}_{1} \leq i\left( {b}_{0}\right) \leftrightarrow \# \left( {b}_{1}\righ... | Proof. 1-4 follow from the definition of \( \# \) . (Note that \( {\mathbf{B}}_{0} \) and \( {\mathbf{B}}_{1} \) are completed and so is \( i \) .)\n\n5. \( i\left( {b}_{0}\right) \cdot {b}_{1} \leq i\left( {{b}_{0} \cdot \# \left( {b}_{1}\right) }\right) \; \) by 1 .\n\n\( \# \left( {i\left( {b}_{0}\right) \cdot {b}_{... | Yes |
Theorem 22.12. If\n\n\\[ \nf : {\\mathbf{P}}_{1}\\xrightarrow[]{\\text{ onto }}{\\mathbf{P}}_{0} \n\\]\n\nis open and continuous and induces \\( i \\), then\n\n\\[ \n\\left( {\\forall {b}_{1} \\in {P}_{1}}\\right) \\left\\lbrack {f\\left( {b}_{1}\\right) = \\# \\left( {b}_{1}\\right) }\\right\\rbrack \n\\]\n\ni.e., \\(... | Proof. There are two complete monomorphisms:\n\n\\[ \nj : {\\mathbf{B}}_{{\\mathbf{P}}_{0}} \\rightarrow {\\mathbf{B}}_{{\\mathbf{P}}_{1}} \n\\]\n\ninduced by \\( f \\) via Theorem 22.9, and\n\n\\[ \ni : {\\mathbf{B}}_{0} \\rightarrow {\\mathbf{B}}_{1} \n\\]\n\nThese monomorphisms are related to each other by\n\n\\[ \n... | Yes |
Theorem 22.13. Let \( {\mathbf{B}}_{0},{\mathbf{B}}_{1},{\mathbf{B}}_{2} \) be complete Boolean algebras, let\n\n\[ \n{i}_{1} : {\mathbf{B}}_{0} \rightarrow {\mathbf{B}}_{1} \n\]\n\n\[ \n{i}_{2} : {\mathbf{B}}_{1} \rightarrow {\mathbf{B}}_{2} \n\]\n\nbe complete monomorphisms and let \( {\# }_{j} \) be the open continu... | Proof. #: \( {\mathbf{B}}_{2} \rightarrow {\mathbf{B}}_{0} \) is open, continuous and onto \( {\mathbf{B}}_{0} \) .\n\n\[ \n{\# }^{-1} = {\left( {\# }_{1} \circ {\# }_{2}\right) }^{-1} = {\# }_{2}{}^{-1} \circ {\# }_{1}{}^{-1}, \n\]\n\nhence\n\n\[ \n{\left( {\# }^{-1}\right) }^{a}{b}_{0} = \left( {{i}_{2} \circ {i}_{1}... | Yes |
Theorem 22.17. \( {\mathbf{B}}_{\alpha } \) is a complete subalgebra of \( \mathbf{B} \) . | Proof. Let \( S \subseteq {B}_{\alpha } \) and \( {b}_{0} = \prod S \) in \( {\mathbf{B}}_{\alpha } \), i.e. \( {b}_{0} = \mathop{\prod }\limits^{{\mathbf{B}}_{\alpha }}\{ s \mid s \in S\} \) . Let \( b \in B \) and suppose that \( \left( {\forall x \in S}\right) \left\lbrack {x \geq b}\right\rbrack \) . We would like ... | Yes |
Theorem 22.18. Let \( \kappa \) be a cardinal or \( {On} \) . Let\n\n\[ \left\langle \left\{ {{X}^{\alpha }\left| {\alpha < \kappa }\right| ,\left\{ {{p}_{\alpha \beta } \mid \alpha \leq \beta < \kappa }\right\} }\right\} \right\rangle \]\n\nbe an o.c.o. inverse system and \( X \) be \( \mathop{\lim }\limits_{{\alpha \... | Proof, By Theorem 22.7, it is sufficient to show that \( {B}_{X} \) is a completion of \( \mathop{\bigcup }\limits_{{\alpha < \kappa }}{B}_{\alpha } \) . For this purpose we have to show (i) each \( {B}_{\alpha } \) is a complete subalgebra of \( {B}_{X} \) and (ii) \( \mathop{\bigcup }\limits_{{\alpha < \kappa }}{B}_{... | Yes |
Moreover, if \( {cf}\left( \kappa \right) \geq \lambda \), then \( X \) is homeomorphic to \( \mathop{\lim }\limits_{{\alpha \rightarrow \kappa }}{X}^{\alpha } \) . | Proof. (We prove only the second part leaving the proof of the first part to the reader.) Let \( \Phi : X \rightarrow {X}^{\prime } = \mathop{\lim }\limits_{{\alpha \rightarrow \kappa }}{X}^{\alpha } \) be defined as follows: \( \Phi \left( f\right) \left( \alpha \right) = f \vdash \alpha \) . Then \( \Phi \) is one-to... | No |
Theorem 22.20. Let \( \kappa \) be a cardinal or \( {On} \) . Let\n\n\[ \n\left\langle {\left\{ {{X}^{\alpha } \mid \alpha < \kappa }\right\} ,\left\{ {{p}_{\alpha \beta } \mid \alpha \leq \beta < \kappa }\right\} }\right\rangle \n\]\n\nbe an o.c.o. inverse system and \( X = \mathop{\lim }\limits_{{\alpha \rightarrow \... | Proof. Define \( q \) by\n\n\[ \n\left( {q\left( y\right) }\right) \left( \alpha \right) = {q}_{\alpha }\left( y\right) \n\]\n\nObviously \( q\left( y\right) \in X \) and \( {q}_{\alpha } = {p}_{\alpha } \circ q \) . Define\n\n\[ \n{X}_{0} = \{ q\left( y\right) \mid y \in Y\} . \n\]\n\n1. \( {X}_{0} \) is dense in \( X... | Yes |
Theorem 22.24. Let \( {\mathbf{P}}_{1} = \left\langle {{P}_{1},{ \leq }_{1}}\right\rangle \) and \( {\mathbf{P}}_{0} = \left\langle {{P}_{0},{ \leq }_{0}}\right\rangle \) be partial order structures and \( p : {\mathbf{P}}_{1} \rightarrow {\mathbf{P}}_{0} \) be o.c.o. Then\n\n1. for every \( x, y \in {P}_{1} \)\n\n\[ x... | Proof.\n\n1. Suppose \( x{ \leq }_{1}y \) . Then \( x \in {\left\lbrack y\right\rbrack }_{{\mathbf{P}}_{1}} \subseteq {\left( {p}^{-1}\right) }^{u}{\left\lbrack p\left( y\right) \right\rbrack }_{{\mathbf{P}}_{0}} \) (since \( p \) is continuous and \( {\left\lbrack y\right\rbrack }_{{\mathbf{P}}_{1}} \) is the smallest... | Yes |
Theorem 22.26. Let \( \kappa \) be a cardinal or \( {On} \) . Let \( {\mathbf{P}}_{0} \subseteq {\mathbf{P}}_{1} \subseteq \cdots \subseteq {\mathbf{P}}_{\alpha } \subseteq \cdots \) \( \left( {\alpha < \kappa }\right) \) be a normal limiting system. If \( x, y \in {P}_{\alpha } \) are compatible in \( {\mathbf{P}}_{\b... | Proof. Suppose \( \left( {\exists z \in {P}_{\beta }}\right) \left\lbrack {z \leq x \land z \leq y}\right\rbrack \) . Then \( {p}_{\alpha \beta }\left( z\right) \leq x \) and \( {p}_{\alpha \beta }\left( z\right) \leq y \) . | Yes |
Theorem 22.28. Let \( \kappa \) be a cardinal or \( {On} \) . Let \( {\mathbf{P}}_{0} \subseteq {\mathbf{P}}_{1} \subseteq \cdots \subseteq {\mathbf{P}}_{\alpha } \subseteq \cdots \) \( \left( {\alpha < \kappa }\right) \) be a weakly normal limiting system. If \( \mathop{\bigcup }\limits_{{\alpha < \kappa }}{\mathbf{P}... | Proof. We have to show that \( {4}^{ * } \) follows from \( {4}^{\prime } \) and 5 . For that, let \( x \in {P}_{\alpha }, y \in {P}_{\beta } \), and \( \alpha < \beta < \kappa \) . Suppose \( {p}_{\alpha \beta }\left( y\right) \leq x \) and \( y \nleq x \) . Then by 5 there exists a \( \gamma \geq \beta \) and a \( z ... | Yes |
Theorem 22.29. Let \( \kappa > \omega \) be a regular cardinal or \( {On} \) and\n\n\[ \left\langle {\left\{ {{\mathbf{P}}_{\alpha } \mid \alpha < \kappa }\right\} ,\left\{ {{p}_{\alpha \beta } \mid \alpha \leq \beta < \kappa }\right\} }\right\rangle \]\n\nbe a normal limiting system. If \( {\mathbf{P}}_{\alpha } = \ma... | Proof. For a member \( x \in \mathop{\bigcup }\limits_{{\alpha < \kappa }}{P}_{\alpha } \), define \( \left| x\right| \) to be the least ordinal \( \alpha \) such that \( x \in {P}_{\alpha } \) . Then we have\n\n\[ x \in \mathop{\bigcup }\limits_{{\alpha < \kappa }}{P}_{\alpha } \rightarrow {cf}\left( \left| x\right| \... | Yes |
Theorem 22.30. Under the same conditions as in the preceding theorem, let \( {\mathbf{B}}_{\alpha } \) be the complete Boolean algebra of regular open sets in \( {\mathbf{P}}_{\alpha } \), let \( \mathbf{B} = \) \( \mathop{\bigcup }\limits_{{\alpha < \kappa }}{\mathbf{B}}_{\alpha } \) and let \( \widetilde{\mathbf{B}} ... | Proof. i). Follows from ii), iii), and Theorem 22.29.\n\niii). Since, by Theorem 22.18 \( \widetilde{\mathbf{B}} \cong {\mathbf{B}}_{2} \), it suffices to show that \( {\mathbf{B}}_{1} \cong {\mathbf{B}}_{2} \) . Clearly there exists a projection \( {q}_{\beta } : \mathop{\bigcup }\limits_{{\alpha < \kappa }}{\mathbf{P... | Yes |
Corollary 22.31. Let \( \kappa > \omega \) be a regular cardinal and \( {\mathbf{B}}_{0} \subseteq {\mathbf{B}}_{1} \subseteq \cdots \subseteq \) \( {\mathbf{B}}_{\alpha } \subseteq \cdots \left( {\alpha < \kappa }\right) \) be a direct system of complete Boolean algebras such that \( {\mathbf{B}}_{\alpha } \) is a com... | Proof. Define \( {C}_{\alpha }\left( {\alpha \leq \kappa }\right) \) and \( C \) as follows.\n\ni) \( {C}_{0} = \left\{ {b \in {B}_{0} \mid b > 0}\right\} \).\n\nii) \( {C}_{\alpha + 1} = \left\{ {b \in {B}_{\alpha + 1} \mid {\# }_{\alpha }\left( b\right) \in {C}_{\alpha }}\right\} \).\n\niii) \( {C}_{\alpha } = \matho... | No |
Theorem 23.8. \( {\mathbf{V}}^{\left( \mathbf{B}\right) } \) and \( {\mathbf{V}}^{\left( \widetilde{\mathbf{B}}\right) } \) satisfy the Axiom of Union. | Proof. We have to show that for a given \( u \in {V}^{\left( \mathbf{B}\right) } \) or \( u \in {V}^{\left( \widehat{\mathbf{B}}\right) } \)\n\n\[ \llbracket \left( {\exists v}\right) \left( {\forall x}\right) \left\lbrack {x \in v \rightsquigarrow \left( {\exists y \in u}\right) \left\lbrack {x \in y}\right\rbrack }\r... | Yes |
Theorem 23.9. \( {\mathrm{V}}^{\left( \mathrm{B}\right) } \) satisfies the Axiom of Subsets (Zermelo’s Axiom Schema of Separation). | Proof. Let \( a \in {V}^{\left( \widetilde{\mathbf{B}}\right) } \) . We wish to prove that\n\n\[ \llbracket \left( {\exists v}\right) \left( {\forall y}\right) \left\lbrack {y \in v \leftrightarrow y \in a \land \varphi \left( y\right) }\right\rbrack \rrbracket = \mathbf{1}. \]\n\nDefine \( v \in {V}^{\left( \widetilde... | Yes |
Theorem 23.11. Suppose \( \left\{ {{b}_{\alpha i} \mid \alpha \in {On} \land i < 2}\right\} \subseteq \widetilde{B} \) , 1. \( \left( {\forall \alpha }\right) \left( {\forall \beta }\right) \left\lbrack {\alpha < \beta \rightarrow {b}_{⓴} \leq {b}_{⓪} \land {b}_{⓵} \leq {b}_{⓫}}\right\rbrack \) and 2. \( \mathop{\prod ... | Proof. (By contradiction.) Suppose \( b = \mathop{\prod }\limits_{\alpha }\left( {{b}_{⓪} + {b}_{⓫}}\right) > 0 \) . Since \( \mathop{\prod }\limits_{\alpha }{b}_{⓪} = \mathbf{0},{b}_{⓪} \ngeq b \) for some \( \alpha \) . Let \( p = b - {b}_{⓪} \) . Then \( \mathbf{0} < p \leq b \) and \( p \cdot {b}_{⓪} = \mathbf{0} \... | Yes |
Theorem 23.12. Let \( \widetilde{\mathbf{B}} \) be the completion of \( \mathbf{B} \) . For \( b \in \widetilde{B} \) define\n\n\[ \n{b}_{\alpha } = \sup \left\{ {{b}^{\prime } \mid {b}^{\prime } \in B \cap R\left( \alpha \right) \land {b}^{\prime } \leq b}\right\} .\n\]\n\nThen\n\n1. \( {b}_{\alpha } \leq b \) .\n\n2.... | Proof. 1 and 2 are obvious.\n\n3. Suppose \( \mathop{\sum }\limits_{\alpha }{b}_{\alpha } < b \) . Since \( \widetilde{\mathbf{B}} \) is the completion of \( \mathbf{B} \) ,\n\n\[ \n\left( {\exists p \in B}\right) \left\lbrack {\mathbf{0} < p \leq b - \mathop{\sum }\limits_{{\alpha \in {On}}}{b}_{\alpha }}\right\rbrack... | No |
Theorem 23.15. Suppose \( \mathop{\lim }\limits_{{\alpha \in {On}}}{b}_{\alpha } = b \) and \( \mathop{\lim }\limits_{{\alpha \in {On}}}{b}_{\alpha }^{\prime } = {b}^{\prime } \) . Then\n\n1. \( \mathop{\lim }\limits_{{\alpha \in {On}}}\left( {-{b}_{\alpha }}\right) = - b \) .\n\n2. \( \mathop{\lim }\limits_{{\alpha \i... | Proof. 1. Obvious since \( \left( {{b}_{\alpha } \Leftrightarrow b}\right) = \left( {-{b}_{\alpha } \Leftrightarrow - b}\right) \) .\n\n2. \( \left( {{b}_{\alpha } + {b}_{\alpha }^{\prime }}\right) \Leftrightarrow \left( {b + {b}^{\prime }}\right) \)\n\n\[= - \left( {{b}_{\alpha } + {b}_{\alpha }^{\prime }}\right) \cdo... | Yes |
Theorem 23.16. Let \( \widetilde{\mathbf{B}} \) be the completion of \( \mathbf{B} \) . If \( \widetilde{\mathbf{B}} \) satisfies the UCL, then \( \left( {\forall v \in {V}^{\left( \widetilde{\mathbf{B}}\right) }}\right) \left( {\exists \left\{ {{u}_{\alpha } \mid \alpha \in {On}}\right\} \subseteq {V}^{\left( \mathbf{... | Proof. (By induction on the least \( \alpha \) such that \( v \in {V}_{\alpha }{}^{\left( \widetilde{\mathbf{B}}\right) } \) .) As our induction hypothesis, assume\n\n\[ \left( {\forall x \in \mathcal{D}\left( v\right) }\right) \left( {\exists \left\{ {{y}_{\alpha x} \mid \alpha \in {On}}\right\} \subseteq {V}^{\left( ... | Yes |
Corollary 23.17. If \( \widetilde{\mathbf{B}} \) is the completion of \( \mathbf{B} \) and \( \widetilde{\mathbf{B}} \) satisfies the UCL, then\n\n\[ \left( {\forall p \in \widetilde{B}}\right) \lbrack p > \mathbf{0} \rightarrow \left( {\forall v \in {V}^{\left( \widetilde{\mathbf{B}}\right) }}\right) \left( {\exists u... | Proof. For given \( v \) and \( p > 0 \) let\n\n\[ {b}_{\alpha }^{\prime } = \mathop{\prod }\limits_{{\beta \geq \alpha }}\mathop{\prod }\limits_{{x \in \mathcal{D}\left( v\right) }}\left( {{b}_{\beta x} \Leftrightarrow v\left( x\right) }\right) . \]\n\nSince \( \mathop{\sum }\limits_{\alpha }{b}_{\alpha }^{\prime } = ... | Yes |
Theorem 23.18. Suppose that \( \widetilde{\mathbf{B}} \) is the completion of \( \mathbf{B} \) and \( \widetilde{\mathbf{B}} \) satisfies the UCL. Then \( {\mathbf{V}}^{\left( \mathbf{B}\right) } \) is a \( \widetilde{\mathbf{B}} \) -valued elementary substructure of \( {\mathbf{V}}^{\left( \widetilde{\mathbf{B}}\right... | Proof. (By induction on the number of logical symbols in \( \varphi \) .) We consider only the case of a quantifier.\n\nLet \( \varphi \left( {{u}_{1},\ldots ,{u}_{n}}\right) = \left( {\forall x}\right) \psi \left( {x,{u}_{1},\ldots ,{u}_{n}}\right) \) where \( {u}_{1},\ldots ,{u}_{n} \in {V}^{\left( \mathbf{B}\right) ... | Yes |
Theorem 23.21. If B satisfies the s.c.c. then B satisfies the UCL. | Proof. Suppose that there is a family of sequences \( \left\langle {{b}_{\alpha i} \mid \alpha \in {On}}\right\rangle \) for each \( i \in I \), where \( I \) is a set, such that\n\n\[ \left\{ {{b}_{\alpha i} \mid \alpha \in {On} \land i \in I}\right\} \subseteq B, \]\n\n\[ \left( {\forall i \in I}\right) \left( {\fora... | Yes |
Theorem 23.22. Let \( \widetilde{\mathbf{B}} \) be the completion of \( \mathbf{B} \) . If \( \widetilde{\mathbf{B}} \) satisfies the UCL, then \( {\mathbf{V}}^{\left( \mathbf{B}\right) } \) satisfies the Axiom of Replacement. | Proof. As in the proof of Theorem 9.25 we have to show that\n\n(i) \( \llbracket \left( {\forall x}\right) \left( {\exists y}\right) {\varphi }^{\prime }\left( {x, y}\right) \rrbracket = 1 \)\n\nimplies\n\n(ii) \( \left( {\forall a \in {V}^{\left( \mathbf{B}\right) }}\right) \left\lbrack {\lbrack \left( {\exists v}\rig... | Yes |
Theorem 23.23. \( \llbracket {AC}\rrbracket = 1 \) in \( {\mathrm{V}}^{\left( \mathrm{B}\right) } \) . | Proof. In most cases \( \mathbf{B} \) satisfies the additional requirement that \( {\mathbf{V}}^{\left( \mathbf{B}\right) } \) is a model of \( {ZF} \) . In this case we can prove the \( {AC} \) in \( {\mathbf{V}}^{\left( \mathbf{B}\right) } \) by a forcing argument just as in Theorem 14.25 with suitable modifications ... | Yes |
Theorem 23.24. Suppose that \( {\mathbf{V}}^{\left( \mathbf{B}\right) } \) is a model of \( {ZF} \) and assume \( {ACH} \) . Then \( {ACH} \) is \( \mathbf{B} \) -valid in \( {\mathbf{V}}^{\left( \mathbf{B}\right) } \) . | Proof. (The proof shows how to apply forcing arguments in the case where \( \mathbf{B} \) is a class.) Let \( \mathbf{M} \) be a countable transitive structure such that \( \left\langle {\mathbf{M},{\mathbf{B}}^{\mathbf{M}},{H}^{\mathbf{M}}}\right\rangle \) is an elementary substructure of \( \langle V,\mathbf{B}, H{\r... | Yes |
Theorem 23.26. There is a Boolean algebra \( \widetilde{\mathbf{B}} \) which satisfies the UCL (even more, \( \widetilde{\mathbf{B}} \) satisfies the c.c.c.), but \( {\mathbf{V}}^{\left( \widetilde{\mathbf{B}}\right) } \) does not satisfy the Axiom of Powers. | Proof. To prove Theorem 11.11 we used the partial order structure \( {\mathbf{P}}_{\alpha } = \left\langle {{P}_{\alpha }, \leq }\right\rangle \) where\n\n\[ \n{P}_{\alpha } = \{ p \mid \left( {\exists d}\right) \left\lbrack {d \subseteq \alpha \times \omega \land \bar{d} < \omega \land p : d \rightarrow 2}\right\rbrac... | Yes |
Theorem 23.30. Suppose that \( \widetilde{\mathbf{B}} \) satisfies the UCL and \( \mathbf{B} \) satisfies the SL, where \( \widetilde{\mathbf{B}} \) is the completion of \( \mathbf{B} \). Then \( {\mathbf{V}}^{\left( \mathbf{B}\right) } \) and \( {\mathbf{V}}^{\left( \widetilde{\mathbf{B}}\right) } \) satisfy the Axiom... | Proof. Again let \( \mathbf{M} \) be a countable transitive structure such that \( \left\langle {\mathbf{M},{\mathbf{B}}^{\mathbf{M}}}\right\rangle \) is an elementary substructure of \( \langle V,\mathbf{B}\rangle \) with respect to the language \( \mathcal{L} \) * of \( {\mathbf{V}}^{\left( \mathbf{B}\right) } \), le... | No |
Theorem 23.32. If \( \mathbf{B} \) satisfies the c.c.c. and \( {cf}\left( {\aleph }_{\beta }\right) > {\aleph }_{0} \), then \( \mathbf{B} \) is \( \left( {{\aleph }_{\alpha },{\aleph }_{\beta }}\right) \) -splitable. | Proof. If \( \mathbf{B} \) satisfies the c.c.c. and \( {cf}\left( {\aleph }_{\beta }\right) > {\aleph }_{0} \), then\n\n\[ b = \mathop{\sum }\limits_{{j < {\aleph }_{\beta }}}{b}_{ij} = \mathop{\sum }\limits_{{j < \gamma }}{b}_{ij}\;\text{ for some }\;\gamma < {\aleph }_{\beta }.\]\n\nTherefore we can simply take \( \b... | No |
Corollary 23.34. If \( {\mathbf{V}}^{\left( \mathbf{B}\right) } \) satisfies the axioms of \( {ZF} + {AC} \) and \( \mathbf{B} \) is \( \left( {{\aleph }_{\alpha },{\aleph }_{\beta }}\right) \) -splitable for all \( {\aleph }_{\alpha },{\aleph }_{\beta } \) such that \( {cf}\left( {\aleph }_{\beta }\right) > {\aleph }_... | Proof. One can easily prove that if for all \( \alpha \) and \( \beta \) \n\n\[ \n{cf}\left( {\aleph }_{\beta }\right) > {\aleph }_{\alpha } \rightarrow \left\lbrack {{cf}{\left( {\aleph }_{\beta }\right) }^{ \vee } > {\left( {\aleph }_{\alpha }\right) }^{ \vee }}\right\rbrack = 1 \n\] \n\nthen \( \left( {\forall \alph... | Yes |
Theorem 23.36. If \( \mathbf{B} \) satisfies the \( \left( {{\omega }_{\alpha },{\omega }_{\beta }}\right) \) -WDL, then \( \mathbf{B} \) is \( \left( {{\omega }_{\alpha },{\omega }_{\beta }}\right) \) - splitable. | Proof. Let \( \mathbf{0} < r \leq b \land \left( {\forall \xi < {\omega }_{\alpha }}\right) \left\lbrack {b = \mathop{\sum }\limits_{{\eta < {\omega }_{\beta }}}{b}_{\xi \eta }}\right\rbrack \) . Then, by the \( \left( {{\omega }_{\alpha },{\omega }_{\beta }}\right) \) -WDL,\n\n\[ b = \mathop{\prod }\limits_{{\xi < {\o... | Yes |
Corollary 23.37. Suppose \( {cf}\left( {\aleph }_{\beta }\right) > {\aleph }_{\alpha } \) and \( {\mathbf{V}}^{\left( \mathbf{B}\right) } \) satisfies the axioms of \( {ZF} + {AC} \) . Then the following conditions are equivalent.\n\n(i) B satisfies the \( \left( {{\omega }_{\alpha },{\omega }_{\beta }}\right) \) -WDL.... | Proof. We can prove (i) \( \leftrightarrow \) (iii) in the same way we proved Theorem 20.3. By Theorem 23.36,(i) \( \rightarrow \) (ii); and by Theorem 23.32,(ii) \( \rightarrow \) (iii). | No |
Theorem 23.42. Suppose \( b \in B,\left\{ {{b}_{j} \mid j \in J}\right\} \subseteq B, b = \mathop{\sum }\limits_{{j \in J}}{b}_{j} \), where \( J \) may be a proper class, and \( {b}^{\prime } \in \widetilde{P} \) . Then \[ \left( {\exists p \in \Gamma }\right) \left( {\exists q \in \Delta }\right) \left\lbrack {p \cdo... | Proof. Case 1: \( {b}^{\prime } \cdot b > \mathbf{0} \) . Then \( {b}^{\prime } \cdot {b}_{j} > \mathbf{0} \) for some \( j \in J \) . Hence \[ \left( {\exists p \in \Gamma }\right) \left( {\exists q \in \Delta }\right) \left\lbrack {p \cdot q \leq {b}^{\prime } \cdot {b}_{j}}\right\rbrack \] since \( P \) is dense in ... | Yes |
Theorem 23.45. Let \( \bar{I} \leq {\aleph }_{\alpha } \) and \( {\aleph }_{\alpha } \) be regular. Then \( \mathbf{B} \) (the Boolean algebra of regular open sets in \( \mathbf{P} \) ) satisfies the I-SL. | Proof. We claim that \( {\mathbf{B}}_{0} \) (the Boolean algebra of regular open sets in \( {\mathbf{P}}_{0} \) ) satisfies the condition for \( {\mathbf{B}}_{0} \) in the I-SL. Let \( \left\{ {{b}_{i} \mid i \in I}\right\} \subseteq B \) and \( r > \mathbf{0} \) . By Theorem 23.44, there exists \( \bar{r} \leq r \) an... | Yes |
Theorem 24.2.\n\n5. \( \left\{ {{q}_{\beta } \mid \beta < {\aleph }_{\alpha }}\right\} \subseteq {\Delta }_{\alpha } \land \left( {\forall \beta }\right) \left( {\forall \delta }\right) \left\lbrack {\beta \leq \delta < {\aleph }_{\alpha } \rightarrow {q}_{\delta } \leq {q}_{\beta }}\right\rbrack \n\n\[ \rightarrow q =... | Proof. 1-4 are obvious from the definitions.\n\n5. We need only prove that \( q = \mathop{\bigcup }\limits_{{\beta < {\aleph }_{\alpha }}}{q}_{\beta } \in P \) . Let \( {q}^{\gamma } = \mathop{\bigcup }\limits_{{\beta < {\aleph }_{\alpha }}}{q}_{\beta }{}^{\gamma } \) . Then 1.i-1.iii of Definition 24.1 are satisfied. ... | Yes |
Theorem 24.4. The map \( j : {\Gamma }_{\beta } \rightarrow {\Gamma }_{\alpha } \) has the following elementary properties:\n\n1. \( q \leq p \rightarrow j\left( q\right) \leq j\left( p\right) \) .\n\n2. \( {j}^{\alpha }{\left\lbrack p\right\rbrack }_{{\mathbf{P}}_{\beta }} = {\left\lbrack j\left( p\right) \right\rbrac... | Proof. 1. Obvious.\n\n2. \( {j}^{\alpha }{\left\lbrack p\right\rbrack }_{{\mathbf{r}}_{\beta }} \subseteq {\left\lbrack j\left( p\right) \right\rbrack }_{{\mathbf{r}}_{\alpha }} \) is obvious from 1. Now take \( q \in {\left\lbrack j\left( p\right) \right\rbrack }_{{\mathbf{r}}_{\alpha }} \) . Then \( q \cup p \in {\Ga... | No |
Theorem 24.5. Let \( p \in {\Gamma }_{\beta } \) and \( \beta > \alpha \) . Then \( \# \left( {\left\lbrack p\right\rbrack }_{{\mathbf{P}}_{\beta }}\right) = {\left\lbrack j\left( p\right) \right\rbrack }_{{\mathbf{P}}_{\alpha }} \) . | Proof. By the definition of #. (See Remark following Theorem 22.9.)\n\n\[ \# \left( {\left\lbrack p\right\rbrack }_{{\mathbf{P}}_{\beta }}\right) = \inf \left\{ {b \in {\mathbf{B}}_{\alpha } \mid {\left\lbrack p\right\rbrack }_{{\mathbf{P}}_{\beta }} \leq i\left( b\right) }\right\} \]\n\n\[ = \inf \left\{ {b \in {\math... | Yes |
Lemma 24.6. Assume \( {h}_{0}\left( \left\lbrack {u \subseteq {\left( {\aleph }_{\alpha }\right) }^{ \vee }}\right\rbrack \right) = 1 \) for some \( u \in {\left( {V}^{\left( \mathbf{B}\right) }\right) }^{\mathbf{M}} \) . Then there exists a \( p \in P \) and a \( \Lambda \subseteq {\Gamma }_{\alpha } \) such that \( {... | Proof. Applying Theorem 23.44 in \( \left\langle {\mathbf{M},{\mathbf{B}}^{\mathbf{M}}}\right\rangle \) to any \( q \leq \left\lbrack {u \subseteq {\left( {\mathbf{N}}_{\alpha }\right) }^{ \sim }}\right\rbrack \) , with \( r = q, I = {\aleph }_{\alpha },{b}_{\gamma } = \llbracket \check{\gamma } \in u\rrbracket \) for ... | Yes |
Runge Approximation Theorem | 196ff., 200 | No |
Proposition 1.3. \( d \) is an antiderivation, i.e., \[ d\left( {\tau \cdot \omega }\right) = \left( {d\tau }\right) \cdot \omega + {\left( -1\right) }^{\deg \tau }\tau \cdot {d\omega }.\] | Proof. By linearity it suffices to check on monomials \[ \tau = {f}_{I}d{x}_{I},\omega = {g}_{J}d{x}_{J}. \] \[ d\left( {\tau \cdot \omega }\right) = d\left( {{f}_{I}{g}_{J}}\right) d{x}_{I}d{x}_{J} = \left( {d{f}_{I}}\right) {g}_{J}d{x}_{I}d{x}_{J} + {f}_{I}d{g}_{J}d{x}_{I}d{x}_{J} \] \[ = \left( {d\tau }\right) \cdot... | Yes |
Proposition 1.4. \( {d}^{2} = 0 \) . | Proof. This is basically a consequence of the fact that the mixed partials are equal. On functions,\n\n\[ \n{d}^{2}f = d\left( {\mathop{\sum }\limits_{i}\frac{\partial f}{\partial {x}_{i}}d{x}_{i}}\right) = \mathop{\sum }\limits_{{i, j}}\frac{{\partial }^{2}f}{\partial {x}_{j}\partial {x}_{i}}d{x}_{j}d{x}_{i}.\n\]\n\nH... | Yes |
Proposition 2.1. With the above definition of the pullback map \( {f}^{ * } \) on forms, \( {f}^{ * } \) commutes with \( d \) . | Proof. The proof is essentially an application of the chain rule.\n\n\[ d{f}^{ * }\left( {{g}_{I}d{y}_{{i}_{1}}\ldots d{y}_{{i}_{q}}}\right) = d\left( {\left( {{g}_{I} \circ f}\right) d{f}_{{i}_{1}}\ldots d{f}_{{i}_{q}}}\right) = d\left( {{g}_{I} \circ f}\right) d{f}_{{i}_{1}}\ldots d{f}_{{i}_{q}}. \]\n\n\[ {f}^{ * }d\... | Yes |
Proposition 2.3. The Mayer-Vietoris sequence is exact. | The exactness is clear except at the last step. We first consider the case of functions on \( M = {\mathbb{R}}^{1} \) . Let \( f \) be a \( {C}^{\infty } \) function on \( U \cap V \) as shown in Figure 2.1. We must write \( f \) as the difference of a function on \( U \) and a function on \( V \) . Let \( \left\{ {{\r... | No |
Example 2.6 (The cohomology of the circle). Cover the circle with two open sets \( U \) and \( V \) as shown in Figure 2.2. The Mayer-Vietoris sequence gives | The difference map \( \delta \) sends \( \left( {\omega ,\tau }\right) \) to \( \left( {\tau - \omega ,\tau - \omega }\right) \), so im \( \delta \) is 1 - dimensional. It follows that ker \( \delta \) is also 1-dimensional. Therefore,\n\n\[ \n{H}^{0}\left( {S}^{1}\right) = \ker \delta = \mathbb{R} \]\n\n\[ \n{H}^{1}\l... | Yes |
Proposition 2.7. The Mayer-Vietoris sequence of forms with compact support\n\n\[ 0 \leftarrow {\Omega }_{c}^{ * }\left( M\right) \leftarrow {\Omega }_{c}^{ * }\left( U\right) \oplus {\Omega }_{c}^{ * }\left( V\right) \leftarrow {\Omega }_{c}^{ * }\left( {U \cap V}\right) \leftarrow 0 \]\n\nis exact. | Proof. This time exactness is easy to check at every step. We do it for the last step. Let \( \omega \) be a form in \( {\Omega }_{c}^{ * }\left( M\right) \) . Then \( \omega \) is the image of \( \left( {{\rho }_{U}\omega ,{\rho }_{V}\omega }\right) \) in \( {\Omega }_{c}^{ * }\left( U\right) \oplus {\Omega }_{c}^{ * ... | Yes |
A manifold \( M \) of dimension \( n \) is orientable if and only if it has a global nowhere vanishing n-form. | Proof. Observe that \( T : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) is orientation-preserving if and only if \( {T}^{ * }d{x}_{1}\ldots d{x}_{n} \) is a positive multiple of \( d{x}_{1}\ldots d{x}_{n} \) at every point.\n\n\( \left( \Leftarrow \right) \) Suppose \( M \) has a global nowhere-vanishing \( n \) -f... | Yes |
Proposition 3.3. The definition of the integral \( {\int }_{M}\tau \) is independent of the oriented atlas \( \left\{ \left( {{U}_{\alpha },{\phi }_{\alpha }}\right) \right\} \) and the partition of unity \( \left\{ {\rho }_{\alpha }\right\} \) . | Proof. Let \( \left\{ {V}_{\beta }\right\} \) be another oriented atlas of \( M \), and \( \left\{ {\chi }_{\beta }\right\} \) a partition of unity subordinate to \( \left\{ {V}_{\beta }\right\} \) . Since \( \mathop{\sum }\limits_{\beta }{\chi }_{\beta } = 1 \) ,\n\n\[ \mathop{\sum }\limits_{\alpha }{\int }_{{U}_{\alp... | Yes |
Lemma 3.4. Let \( T : {\mathbb{H}}^{n} \rightarrow {\mathbb{H}}^{n} \) be a diffeomorphism of the upper half space with everywhere positive Jacobian determinant. \( T \) induces a map \( \bar{T} \) of the boundary of \( {\mathbb{H}}^{n} \) to itself. The induced map \( \bar{T} \), as a diffeomorphism of \( {\mathbb{R}}... | Proof. By the inverse function theorem an interior point of \( {\mathbb{H}}^{n} \) must be the image of an interior point. Hence \( T \) maps the boundary to the boundary. We will check that \( \bar{T} \) has positive Jacobian determinant for \( n = 2 \) ; the general case is similar.\n\nLet \( T \) be given by\n\n\[ {... | Yes |
Corollary 4.1.2 (Homotopy Axiom for de Rham Cohomology). Homotopic maps induce the same map in cohomology. | Proof. Recall that a homotopy between two maps \( f \) and \( g \) from \( M \) to \( N \) is a map \( F : M \times {\mathbb{R}}^{1} \rightarrow N \) such that\n\n\[ \left\{ \begin{array}{lll} F\left( {x, t}\right) = f\left( x\right) & \text{ for } & t \geq 1 \\ F\left( {x, t}\right) = g\left( x\right) & \text{ for } &... | Yes |
Proposition 4.6. \( 1 - {e}_{ * }{\pi }_{ * } = {\left( -1\right) }^{q - 1}\left( {{dK} - {Kd}}\right) \) on \( {\Omega }_{c}^{q}\left( {M \times {\mathbb{R}}^{1}}\right) \). | Proof. On forms of type (I), assuming \( \deg \phi = q \), we have\n\n\[ \left( {1 - {e}_{ * }{\pi }_{ * }}\right) \phi \cdot f = \phi \cdot f \]\n\n\[ \left( {{dK} - {Kd}}\right) \phi \cdot f = - K\left( {{d\phi } \cdot f + {\left( -1\right) }^{q}\phi \frac{\partial f}{\partial x}{dx} + {\left( -1\right) }^{q}\phi \fr... | Yes |
Corollary 4.7.1 (Poincaré Lemma for Compact Supports).\n\n\[ \n{H}_{c}^{ * }\left( {\mathbb{R}}^{n}\right) = \left\{ \begin{array}{ll} \mathbb{R} & \text{ in dimension }n \\ 0 & \text{ otherwise. } \end{array}\right. \n\] | Here the isomorphism \( {H}_{c}^{n}\left( {\mathbb{R}}^{n}\right) \simeq \mathbb{R} \) is given by iterated \( {\pi }_{ * } \), i.e., by integration over \( {\mathbb{R}}^{n} \) .\n\nTo determine a generator for \( {H}_{c}^{n}\left( {\mathbb{R}}^{n}\right) \), we start with the constant function 1 on a point and iterate... | Yes |
Theorem 4.9 (Sard’s Theorem for \( {\mathbb{R}}^{n} \) ). The set of critical values of a smooth map \( f : {\mathbb{R}}^{m} \rightarrow {\mathbb{R}}^{n} \) has measure zero in \( {\mathbb{R}}^{n} \) for any integers \( m \) and \( n \) . | This means that given any \( \varepsilon > 0 \), the set of critical values can be covered by cubes with total volume less than \( \varepsilon \) . Important special cases of this theorem were first published by A. P. Morse [1]. Sard's proof of the general case may be found in Sard [1]. | No |
Proposition 4.10 Let \( f : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) be a proper map. Iff is not surjective, then it has degree 0. | Proof. Since the image of a proper map is closed (why?), if \( f \) misses a point \( q \), it must miss some neighborhood \( U \) of \( q \) . Choose a bump \( n \) -form \( \alpha \) whose support lies in \( U \) . Then \( {f}^{ * }\alpha \equiv 0 \) so that \( \deg f = 0 \) . | No |
Theorem 5.1. Every manifold has a good cover. If the manifold is compact, then the cover may be chosen to be finite. | To prove this theorem we will need a little differential geometry. A Riemannian structure on a manifold \( M \) is a smoothly varying metric \( \langle \) , on the tangent space of \( M \) at each point; it is smoothly varying in the following sense: if \( X \) and \( Y \) are two smooth vector fields on \( M \), then ... | Yes |
Proposition 5.3.1. If the manifold \( M \) has a finite good cover, then its cohomology is finite dimensional. | Proof. From the Mayer-Vietoris sequence\n\n\[ \cdots \rightarrow {H}^{q - 1}\left( {U \cap V}\right) \overset{{d}^{ * }}{ \rightarrow }{H}^{q}\left( {U \cup V}\right) \overset{r}{ \rightarrow }{H}^{q}\left( U\right) \bigoplus {H}^{q}\left( V\right) \rightarrow \cdots \]\n\nwe get\n\n\[ {H}^{q}\left( {U \cup V}\right) \... | Yes |
Lemma 5.6. The two Mayer-Vietoris sequences (2.4) and (2.8) may be paired together to form a sign-commutative diagram \n\nHere sign-commutativity means, for instance,\n\n\[ \n{\int }_{U \cap V}\omega \land {d}_{ * }\ta... | Proof. The first two squares are in fact commutative as is straightforward to check. We will show the sign-commutativity of the third square.\n\nRecall from (2.5) and (2.7) that \( {d}^{ * }\omega \) is a form in \( {H}^{q + 1}\left( {U \cup V}\right) \) such that\n\n\[ \n{\left. {d}^{ * }\omega \right| }_{U} = - d\lef... | Yes |
Theorem 5.11 (Leray-Hirsch). Let \( E \) be a fiber bundle over \( M \) with fiber \( F \) . Suppose \( M \) has a finite good cover. If there are global cohomology classes \( {e}_{1},\ldots ,{e}_{r} \) on \( E \) which when restricted to each fiber freely generate the cohomology of the fiber, then \( {H}^{ * }\left( E... | \[ {H}^{ * }\left( E\right) \simeq {H}^{ * }\left( M\right) \otimes \mathbb{R}\left\{ {{e}_{1},\ldots ,{e}_{r}}\right\} \simeq {H}^{ * }\left( M\right) \otimes {H}^{ * }\left( F\right) . \] | Yes |
Lemma 6.1. If the cocycle \( \left\{ {g}_{\alpha \beta }^{\prime }\right\} \) comes from another trivialization \( \left\{ {\phi }_{\alpha }^{\prime }\right\} \), then there exist maps \( {\lambda }_{\alpha } : {U}_{\alpha } \rightarrow {GL}\left( {n,\mathbb{R}}\right) \) such that\n\n\[ \n{g}_{\alpha \beta } = {\lambd... | Proof. The two trivializations differ by a nonsingular transformation of \( {\mathbb{R}}^{n} \) at each point:\n\n\[ \n{\phi }_{\alpha } = {\lambda }_{\alpha }{\phi }_{\alpha }^{\prime }\;,\;{\lambda }_{\alpha } : {U}_{\alpha } \rightarrow {GL}\left( {n,\mathbb{R}}\right) .\n\]\n\nTherefore,\n\n\[ \n{g}_{\alpha \beta }... | Yes |
Theorem 6.8 (Homotopy Property of Vector Bundles). Assume \( Y \) to be a compact manifold. If \( {f}_{0} \) and \( {f}_{1} \) are homotopic maps from \( Y \) to a manifold \( X \) and \( E \) is a vector bundle on \( X \), then \( {f}_{0}^{-1}E \) is isomorphic to \( {f}_{1}^{-1}E \), i.e., homotopic maps induce isomo... | Proof. The problem of constructing an isomorphism between two vector bundles \( V \) and \( W \) of rank \( k \) over a space \( B \) may be turned into a problem in cross-sectioning a fiber bundle over \( B \), as follows. Recall that \( \operatorname{Hom}\left( {V, W}\right) = {V}^{ * } \otimes W \) is a vector bundl... | Yes |
Corollary 6.9. A vector bundle over a contractible manifold is trivial. | Proof. Let \( E \) be a vector bundle over \( M \) and let \( f \) and \( g \) be maps\n\n\[ M\underset{g}{\overset{f}{ \rightleftarrows }}\text{ point } \]\n\nsuch that \( g \circ f \) is homotopic to the identity \( {1}_{M} \) . By the homotopy property of vector bundles\n\n\[ E \simeq {\left( g \circ f\right) }^{-1}... | Yes |
Lemma 6.12. An orientable vector bundle \( E \) over an orientable manifold \( M \) is an orientable manifold. | Proof. This follows from the fact that if \( \left\{ \left( {{U}_{\alpha },{\psi }_{\alpha }}\right) \right\} \) is an oriented atlas for \( M \) with transition functions \( {h}_{\alpha \beta } = {\psi }_{\alpha } \circ {\psi }_{\beta }^{-1} \) and\n\n\[ \n{\left. {\phi }_{\alpha } : E\right| }_{{U}_{\alpha }} \simeq ... | Yes |
Proposition 6.14.1. Integration along the fiber \( {\pi }_{ * } \) commutes with exterior differentiation \( d \) . | Proof. Let \( \left\{ \left( {{U}_{\alpha },{\phi }_{\alpha }}\right) \right\} \) be a trivialization for \( E,\left\{ {\rho }_{\alpha }\right\} \) a partition of unity subordinate to \( \left\{ {U}_{\alpha }\right\} \), and \( \omega \) a form in \( {\Omega }_{cv}^{ * }\left( E\right) \) . Since \( \omega = \sum {\rho... | Yes |
Proposition 6.15 (Projection Formula). (a) Let \( \pi : E \rightarrow M \) be an oriented rank \( n \) vector bundle, \( \tau \) a form on \( M \) and \( \omega \) a form on \( E \) with compact support along the fiber. Then\n\n\[ \n{\pi }_{ * }\left( {\left( {{\pi }^{ * }\tau }\right) \cdot \omega }\right) = \tau \cdo... | Proof. (a) Since two forms are the same if or only if they are the same locally, we may assume that \( E \) is the product bundle \( M \times {\mathbb{R}}^{n} \) . If \( \omega \) is a form of type (I), say \( \omega = {\pi }^{ * }\phi \cdot f\left( {x, t}\right) d{t}_{{i}_{1}}\ldots d{t}_{{i}_{r}} \), where \( r < n \... | Yes |
Proposition 6.16 (Poincaré Lemma for Compact Vertical Supports). Integration along the fiber defines an isomorphism\n\n\[ \n{\pi }_{ * } : {H}_{cv}^{ * }\left( {M \times {\mathbb{R}}^{n}}\right) \rightarrow {H}^{* - n}\left( M\right) .\n\] | This is a special case of\n\nTheorem 6.17 (Thom Isom | No |
Theorem 6.17 (Thom Isomorphism). If the vector bundle \( \pi : E \rightarrow M \) over a manifold \( M \) of finite type is orientable, then\n\n\[ \n{H}_{cv}^{ * }\left( E\right) \simeq {H}^{* - n}\left( M\right) \n\]\n\nwhere \( n \) is the rank of \( E \) . | Proof. Let \( U \) and \( V \) be open subsets of \( M \) . Using a partition of unity from the base \( M \) we see that\n\n\[ \n0 \rightarrow {\Omega }_{cv}^{ * }\left( {E{|}_{U\; \cup \;V}}\right) \rightarrow {\Omega }_{cv}^{ * }\left( {E{|}_{U}}\right) \oplus {\Omega }_{cv}^{ * }\left( {E{|}_{V}}\right) \rightarrow ... | Yes |
Proposition 6.18. The Thom class \( \Phi \) on a rank \( n \) oriented vector bundle \( E \) can be uniquely characterized as the cohomology class in \( {H}_{cv}^{n}\left( E\right) \) which restricts to the generator of \( {H}_{c}^{n}\left( F\right) \) on each fiber \( F \) . | Proof. Since \( {\pi }_{ * }\Phi = 1,{\left. \Phi \right| }_{\text{fiber }} \) is a bump form on the fiber with total integral 1. Conversely if \( {\Phi }^{\prime } \) in \( {H}_{cv}^{n}\left( E\right) \) restricts to a generator on each fiber, then\n\n\[ \n{\pi }_{ * }\left( {\left( {{\pi }^{ * }\omega }\right) \land ... | Yes |
Proposition 6.19. If \( E \) and \( F \) are two oriented vector bundles over a manifold \( M \), and \( {\pi }_{1} \) and \( {\pi }_{2} \) are the projections then the Thom class of \( E \oplus F \) is \( \Phi \left( {E \oplus F}\right) = {\pi }_{1}^{ * }\Phi \left( E\right) \land {\pi }_{2}^{ * }\Phi \left( F\right) ... | Proof. Let \( m = \operatorname{rank}E \) and \( n = \operatorname{rank}F \) . Then \( {\pi }_{1}^{ * }\Phi \left( E\right) \land {\pi }_{2}^{ * }\Phi \left( F\right) \) is a class in \( {H}_{cv}^{m + n}\left( {E \oplus F}\right) \) whose restriction to each fiber is a generator of the compact cohomology of the fiber, ... | Yes |
The Euler class is functorial, i.e., if \( f : N \rightarrow M \) is a \( {C}^{\infty } \) map and \( E \) is a rank 2 oriented vector bundle over \( M \), then\n\n\[ e\left( {{f}^{-1}E}\right) = {f}^{ * }e\left( E\right) . \] | Proof. Since the transition functions of \( {f}^{-1}E \) are \( {f}^{ * }{g}_{\alpha \beta } \), the proposition is an immediate consequence of (6.38). | No |
Proposition 6.41. The pullback of the Thom class to \( M \) by the zero section is the Euler class. | Let \( \left\{ {U}_{\alpha }\right\} \) be a trivializing cover for \( E,\left\{ {\rho }_{\alpha }\right\} \) a partition of unity subordinate to \( \left\{ {U}_{\alpha }\right\} \), and \( {g}_{\alpha \beta } \) the transition functions for \( E \) . Since\n\n\[ \psi = \frac{d{\theta }_{\alpha }}{2\pi } - {\pi }^{ * }... | No |
Proposition 7.2. The twisted cohomology is invariant under the refinement of open covers. More precisely, let \( {\left\{ \left( {U}_{\alpha },{\phi }_{\alpha }\right) \right\} }_{\alpha \in I} \) be a locally constant trivialization for \( E \) . Suppose \( {\left\{ {V}_{\beta }\right\} }_{\beta \in J} \) is a refinem... | Proof. Since the definition of the differential operator on a twisted complex is local, and \( \phi \) and \( \psi \) agree on the open cover \( \left\{ {V}_{\beta }\right\} \), we have \( {d}_{\phi } = {d}_{\psi } \) . Therefore the two complexes \( {\Omega }_{\phi }^{ * }\left( {M, E}\right) \) and \( {\Omega }_{\psi... | Yes |
Proposition 7.4. If \( E \) is a trivial rank \( n \) vector bundle over a manifold \( M \), with \( \phi \) a trivialization of \( E \) given by \( n \) global sections, then\n\n\[ \n{H}_{\phi }^{ * }\left( {M, E}\right) = {H}^{ * }\left( {M,{\mathbb{R}}^{n}}\right) = {\bigoplus }_{i = 1}^{n}{H}^{ * }\left( M\right) .... | Proof. Let \( {e}_{1},\ldots ,{e}_{n} \) be the \( n \) global sections corresponding to the standard basis of \( {\mathbb{R}}^{n} \) . Then every element in \( {\Omega }^{ * }\left( {M, E}\right) \) can be written uniquely as \( \sum {\omega }_{i} \otimes {e}_{i} \), where \( {\omega }_{i} \in {\Omega }^{ * }\left( M\... | Yes |
Proposition 7.5. If \( {\phi }^{\prime } \) and \( {\psi }^{\prime } \) are two trivializations for \( L \) induced from two atlases \( \phi \) and \( \psi \) on \( M \), then the two twisted complexes \( {\Omega }_{\phi }^{ * }\left( {M, L}\right) \) and \( {\Omega }_{\psi }^{ * }(M \) , \( L \) ) are isomorphic and s... | Proof. By going to a common refinement we may assume that the two atlases \( \phi \) and \( \psi \) have the same open cover. Thus on each \( {U}_{\alpha } \) there are two sets of coordinate functions, \( {\phi }_{\alpha } \) and \( {\psi }_{\alpha } \) (Figure 7.3.).\n\n![6c4fe03b-53bd-4737-8b90-17e75293f241_95_0.jpg... | Yes |
Theorem 7.7 (Stokes’ Theorem for Densities). On any manifold \( M \) of dimension \( n \), orientable or not, if \( \omega \in {\Omega }_{c}^{n - 1}\left( {M, L}\right) \), then\n\n\[{\int }_{M}{d\omega } = 0\] | The proof is essentially the same as (3.5). | No |
Theorem 7.8 (Poincaré Duality). On a manifold \( M \) of dimension \( n \) with a finite good cover, there are nondegenerate pairings\n\n\[ \n{H}^{q}\left( M\right) { \otimes }_{\mathbb{R}}{H}_{c}^{n - q}\left( {M, L}\right) \rightarrow \mathbb{R} \]\n\nand\n\n\[ \n{H}_{c}^{q}\left( M\right) { \otimes }_{\mathbb{R}}{H}... | Proof. By tensoring the Mayer-Vietoris sequences (2.2) and (2.7) with \( \Gamma \left( {M, L}\right) \) we obtain the corresponding Mayer-Vietoris sequences for twisted cohomology. The Mayer-Vietoris argument for Poincaré duality on an orientable manifold then carries over word for word. | No |
Corollary 7.8.1. Let \( M \) be a connected manifold of dimension \( n \) having a finite good cover. Then\n\n\[ \n{H}^{n}\left( M\right) = \left\{ \begin{array}{ll} \mathbb{R} & \text{ if }M\text{ is compact orientable } \\ 0 & \text{ otherwise. } \end{array}\right.\n\] | Proof. By Poincaré duality, \( {H}^{n}\left( M\right) = {H}_{c}^{0}\left( {M, L}\right) \) . Let \( \left\{ {U}_{\alpha }\right\} \) be a coordinate open cover for \( M \) . An element of \( {H}_{c}^{0}\left( {M, L}\right) \) is given by a collection of constants \( {f}_{\alpha } \) on \( {U}_{\alpha } \) satisfying\n\... | Yes |
Theorem 8.1. The double complex \( {C}^{ * }\left( {\mathfrak{U},{\Omega }^{ * }}\right) \) computes the de Rham cohomology of \( M \) : | Proof. In one direction there is the natural map\n\n\[ r : {\Omega }^{ * }\left( M\right) \rightarrow {\Omega }^{ * }\left( U\right) \oplus {\Omega }^{ * }\left( V\right) \subset {C}^{ * }\left( {\mathfrak{U},{\Omega }^{ * }}\right) \]\n\ngiven by the restriction of forms. Our first observation is that \( r \) is a cha... | Yes |
Proposition 8.3. \( {\delta }^{2} = 0 \) . | Proof. Basically this is true because in \( {\left( {\delta }^{2}\omega \right) }_{{\alpha }_{0}\ldots {\alpha }_{p + 2}} \) we omit two indices \( {\alpha }_{i},{\alpha }_{j} \) twice with opposite signs. To be precise,\n\n\[ \n{\left( {\delta }^{2}\omega \right) }_{{\alpha }_{0}\ldots {\alpha }_{p + 2}} = \sum {\left... | Yes |
Proposition 8.5. (The Generalized Mayer-Vietoris Sequence). The sequence\n\n\\[ \n0 \rightarrow {\\Omega }^{ * }\\left( M\\right) \\overset{r}{ \\rightarrow }\\prod {\\Omega }^{ * }\\left( {U}_{{\\alpha }_{0}}\\right) \\overset{\\delta }{ \\rightarrow }\\prod {\\Omega }^{ * }\\left( {U}_{{\\alpha }_{0}{\\alpha }_{1}}\\... | Proof. Clearly \\( {\\Omega }^{ * }\\left( M\\right) \\) is the kernel of the first \\( \\delta \\) since an element of \\( \\prod {\\Omega }^{ * }\\left( {U}_{{\\alpha }_{0}}\\right) \\) is a global form on \\( M \\) if and only if its components agree on the overlaps.\n\nNow let \\( \\left\{ {\\rho }_{\\alpha }\\righ... | Yes |
Proposition 8.8 (Generalized Mayer-Vietoris Principle). The double complex \( {C}^{ * }\left( {\mathfrak{U},{\Omega }^{ * }}\right) \) computes the de Rham cohomology of \( M \) ; more precisely, the restriction map \( r : {\Omega }^{ * }\left( M\right) \rightarrow {C}^{ * }\left( {\mathfrak{U},{\Omega }^{ * }}\right) ... | Proof. Since \( {Dr} = \left( {\delta + d}\right) r = {dr} = {rd}, r \) is a chain map, and so it induces a map \( {r}^{ * } \) in cohomology.\n\nStep 1. \( {r}^{ * } \) is surjective.\n\n\n\nLet \( \phi \) be a cocy... | Yes |
Theorem 8.9. If \( \mathfrak{U} \) is a good cover of the manifold \( M \), then the de Rham cohomology of \( M \) is isomorphic to the Čech cohomology of the good cover\n\n\[ {H}_{DR}^{ * }\left( M\right) \simeq {H}^{ * }\left( {\mathfrak{U},\mathbb{R}}\right) \] | Let us recapitulate here what has transpired so far. First, the basic sequence of inclusions\n\n\[ M \leftarrow {U}_{\alpha } \leftleftarrows {U}_{\alpha \beta } \leftleftarrows {U}_{\alpha \beta \gamma } \leftleftarrows \cdots \]\n\ngives rise to the diagram\n\n ,\n\n\[ \n\delta {\left( {D}^{\prime \prime }K\right) }^{i} = {\left( {D}^{\prime \prime }K\right) }^{i}\delta - {\left( {D}^{\prime \prime }K\right) }^{i - 1}{D}^{\prime \prime }. \n\] | Proof of Lemma 9.6. Since \( \delta \) anticommutes with \( {D}^{\prime \prime } \) and since \( {\delta K} + {K\delta } = 1 \)\n\n\[ \n\delta \left( {{D}^{\prime \prime }K}\right) {\left( {D}^{\prime \prime }K\right) }^{i - 1} = - {D}^{\prime \prime }{\delta K}{\left( {D}^{\prime \prime }K\right) }^{i - 1} \n\]\n\n\[ ... | Yes |
Lemma 10.4.1. \( {\phi }^{\# } \) is a chain map, i.e., it commutes with \( \delta \) . | Proof. \( \;\left( {\delta \left( {{\phi }^{\# }\omega }\right) }\right) \left( {V}_{{\beta }_{0}\ldots {\beta }_{q + 1}}\right) = \sum {\left( -1\right) }^{i}\left( {{\phi }^{\# }\omega }\right) \left( {V}_{{\beta }_{0}\ldots {\widehat{\beta }}_{i}\ldots {\beta }_{q + 1}}\right) \)\n\n\[ = \sum {\left( -1\right) }^{i}... | Yes |
Lemma 10.4.2. Given \( \mathfrak{U} = {\left\{ {U}_{\alpha }\right\} }_{\alpha \in I} \) an open cover and \( \mathfrak{B} = {\left\{ {V}_{\beta }\right\} }_{\beta \in J} \) a refinement, if \( \phi \) and \( \psi \) are two refinement maps: \( J \rightarrow I \), then there is a homotopy operator between \( {\phi }^{ ... | Proof. Define \( K : {C}^{q}\left( {\mathfrak{U},\mathcal{F}}\right) \rightarrow {C}^{q - 1}\left( {\mathfrak{V},\mathcal{F}}\right) \) by\n\n\[ \left( {K\omega }\right) \left( {V}_{{\beta }_{0}\ldots {\beta }_{q - 1}}\right) = \sum {\left( -1\right) }^{i}\omega \left( {U}_{\phi \left( {\beta }_{0}\right) \ldots \phi \... | Yes |
Proposition 10.6. Let \( \mathbb{R} \) be the constant presheaf on a manifold \( M \) . Then the Čech cohomology of \( M \) with values in \( \mathbb{R} \) is isomorphic to the de Rham cohomology. | Proof. Since the good covers are cofinal in the set of all covers of \( M \) (Corollary 5.2), we can use only good covers in the direct limit\n\n\[ \n{H}^{ * }\left( {M,\mathbb{R}}\right) = \mathop{\lim }\limits_{u}{H}^{ * }\left( {\mathfrak{U},\mathbb{R}}\right) \n\]\n\nBy Theorem 8.9,\n\n\[ \n{H}^{ * }\left( {\mathfr... | Yes |
Proposition 11.4. A vector bundle \( E \) is orientable if and only if its determinant bundle \( \det E \) is orientable. | Proof. Let \( \left\{ {g}_{\alpha \beta }\right\} \) be the transition functions of \( E \) . Then the transition functions of det \( E \) are \( \left\{ {\det {g}_{\alpha \beta }}\right\} \) . An orthogonal matrix \( {g}_{\alpha \beta } \) assumes values in \( {SO}\left( {n + 1}\right) \) if and only if det \( {g}_{\a... | Yes |
Proposition 11.5. Every vector bundle over a simply connected base space is orientable. | In particular, the tangent bundle of a simply connected manifold is orientable. Since a manifold is orientable if and only if its tangent bundle is (Example 6.3), this gives | No |
Proposition 11.7. For a given orientation \( \left\{ \left\lbrack {\sigma }_{a}\right\rbrack \right\} \) the Euler class is independent of the choice of \( {\sigma }^{j, n - j}, j = 0,\ldots, n \) . | Proof.\n\n\n\nLet \( {\bar{\sigma }}^{0, n} \) be another cochain in \( {C}^{0}\left( {{\pi }^{-1}\mathfrak{U},{\Omega }^{n}}\right) \) which represents the orientation \( \left\{ \left\lbrack {\sigma }_{\alpha }\rig... | Yes |
The Euler class \( e\left( E\right) \) is independent of the choice of the good cover. | Write \( {\varepsilon }_{\mathfrak{U}} \) for the cocycle in \( {H}^{n + 1}\left( {\mathfrak{U},\mathbb{R}}\right) \) which defines the Euler class in terms of the good cover \( \mathfrak{U} \). If a good cover \( \mathfrak{V} \) is a refinement of \( \mathfrak{U} \), then there is a commutative diagram\n\n![6c4fe03b-5... | Yes |
Proposition 11.9. If the oriented sphere bundle \( E \) has a section, then its Euler class vanishes. | Proof. Let \( s \) be a section of \( E \) . It follows from \( \pi \circ s = 1 \) that \( {s}^{ * }{\pi }^{ * } = 1 \) . We saw in the construction of the Euler class that\n\n\[ - {\pi }^{ * }\varepsilon = {D\sigma } \]\n\nfor some \( D \) -cochain \( \sigma \) . Applying \( {s}^{ * } \) to both sides gives\n\n\[ - \v... | Yes |
Proposition 11.14. Let \( \pi : E \rightarrow M \) be a \( \left( {k - 1}\right) \) -sphere bundle over a compact manifold of dimension \( k \) . Suppose the structure group of \( E \) can be reduced to \( O\left( k\right) \) . Then \( E \) has a section over \( M - \left\{ {{x}_{1},\ldots ,{x}_{q}}\right\} \) for some... | Proof. Since the structure group of \( E \) is \( O\left( k\right) \), we can form a Riemannian vector bundle \( {E}^{\prime } \) of rank \( k \) whose unit sphere bundle is \( E \) . A section \( {s}^{\prime } \) of \( {E}^{\prime } \) over \( M \) gives rise to a partial section \( s \) of \( E : s\left( x\right) = {... | Yes |
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