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Theorem 16.2.21 (Lefschetz duality theorem [195, Theorem 29.1]). The biduality functor \( {}^{\land \land } : {\mathrm{{LLC}}}_{K} \rightarrow {\mathrm{{LLC}}}_{K} \) and \( {1}_{{\mathrm{{LLC}}}_{K}} : {\mathrm{{LLC}}}_{K} \rightarrow {\mathrm{{LLC}}}_{K} \) are naturally isomorphic. This induces a duality between the... | As in Pontryagin-van Kampen duality theorem 13.4.17, one proves that, for every locally linearly compact vector space \( V \), the evaluation map \( {\omega }_{V} : V \rightarrow {V}^{\land \land }, v \mapsto {\omega }_{V}\left( v\right) \) , with \( {\omega }_{V}\left( v\right) \left( \chi \right) = \chi \left( v\righ... | Yes |
Corollary 16.2.22. Let \( V \) be a linearly compact vector space over a discrete field \( K \) . Then \( V \) is compact if and only if \( K \) is finite. In particular, \( V \) is a hereditarily disconnected locally compact abelian group whenever \( K \) is finite. | Proof. By Exercise 16.3.16, \( V = \mathop{\prod }\limits_{{i \in I}}{K}_{i} \) with \( {K}_{i} = K \) for all \( i \in I \) . If \( V \) is compact, then each \( {K}_{i} \) is compact as well, hence \( K \) is finite, being compact and discrete. Conversely, if \( K \) is finite, then each \( {K}_{i} \) is compact, so ... | No |
The Pontryagin-van Kampen duality functor, when restricted to \( p \) -torsion discrete abelian groups, gives a duality between \( p \) -torsion discrete abelian groups and abelian pro- \( p \) -groups. | Both categories consist of \( {\mathrm{J}}_{p} \) -modules; moreover, \( \widehat{G} = \operatorname{Hom}\left( {G,\mathbb{T}}\right) = {\operatorname{Hom}}_{{\mathbb{J}}_{p}}\left( {G,\mathbb{Z}\left( {p}^{\infty }\right) }\right) \) for a discrete \( p \) -torsion abelian group \( G \), and \( \widehat{K} = \operator... | Yes |
Corollary 3. An everreturn null equivalence class \( C \) is either empty or infinite. In particular, a finite chain has no everreturn null states. | Let \( C \) be finite nonempty. By the asymptotic passage theorem, \( {P}_{jk}^{n} \rightarrow 0 \) for \( k \in C \) . But \( C \) is closed, so that \( 1 = \mathop{\sum }\limits_{{k \in C}}{P}_{jk}^{n} \rightarrow 0 \) for \( j \in C \), and we reach a contradiction. | Yes |
Corollary 2. A continuous function \( X \) on a compact metric space \( \left( {\Omega ,\rho }\right) \) to a metric space \( \left( {x, d}\right) \) is uniformly continuous. | By definition, \( X \) is uniformly continuous if for every \( \epsilon > 0 \) there exists a \( \delta = \delta \left( \epsilon \right) > 0 \), which depends only upon \( \epsilon \), such that \( d\left( {X\left( \omega \right), X\left( {\omega }^{\prime }\right) }\right) < \epsilon \) for \( \rho \left( {\omega ,{\o... | Yes |
Corollary 1. Let \( {x}_{0} \) be a nonzero point of a normed linear space, and let \( A \) be a closed linear subspace. There exist linear functionals \( f \) , \( {f}^{\prime } \) on the space such that\n\n\[ \parallel f\parallel = 1\\text{ and }f\\left( {x}_{0}\\right) = \\begin{Vmatrix}{x}_{0}\\end{Vmatrix}, \]\n\n... | Set \( f\\left( {a{x}_{0}}\\right) = a\\begin{Vmatrix}{x}_{0}\\end{Vmatrix},{f}^{\prime }\\left( {a{x}_{0} + x}\\right) = {ad}\\left( {{x}_{0}, A}\\right), x \\in A \), and extend. | No |
Corollary 2. If \( X \) is integrable, then \( {\int }_{A}\left| X\right| \rightarrow 0 \) as \( {\mu A} \rightarrow 0 \) . | For, if \( {X}_{n} = X \) or \( n \) according as \( \left| X\right| < n \) or \( \left| X\right| \geqq n \), then \( \int \left| {X}_{n}\right| \uparrow \) \( \int \left| X\right| \), so that, given \( \epsilon > 0 \), there exists an \( {n}_{0} \) such that \( \int \left| X\right| < \) \( \int \left| {X}_{{n}_{0}}\ri... | Yes |
Corollary 1. \( {X}_{n}\overset{\mathrm{r}}{ \rightarrow }X \) implies \( {X}_{n}\overset{{\mathrm{r}}^{\prime }}{ \rightarrow }X \) for \( {r}^{\prime } < r \) . | Set \( {A}_{n} = \left\lbrack {\left| {{X}_{n} - X}\right| \geqq 1}\right\rbrack \) and observe that\n\n\[ \n{\int }_{A}{\left| {X}_{n} - X\right| }^{{r}^{\prime }} = {\int }_{A{A}_{n}}{\left| {X}_{n} - X\right| }^{{r}^{\prime }} \n\]\n\n\[ \n+ {\int }_{A{A}_{n}{}^{c}}{\left| {X}_{n} - X\right| }^{{r}^{\prime }} \leqq ... | No |
Corollary 2. If \( \sup E{\left| {X}_{n}\right| }^{r} = c < \infty \), then \( {X}_{n}\overset{\mathrm{P}}{ \rightarrow }X \) implies \( {X}_{n}\overset{{\mathrm{r}}^{\prime }}{ \rightarrow }X \) for \( {r}^{\prime } < r \) . | Let \( {A}_{n} = \left\lbrack {\left| {X}_{n}\right| \geqq a}\right\rbrack \) and observe that\n\n\[{\int }_{A}{\left| {X}_{n}\right| }^{{r}^{\prime }} = {\int }_{A{A}_{n}}{\left| {X}_{n}\right| }^{{r}^{\prime }} + {\int }_{A{A}_{n}}{\left| {X}_{n}\right| }^{{r}^{\prime }} \leqq c{a}^{{r}^{\prime } - r} + {a}^{r}{PA} <... | Yes |
Corollary 3. If \( \left| {X}_{n}\right| \leqq Y \in {L}_{r} \) for large \( n \), then \( {X}_{n}\overset{\mathrm{P}}{ \rightarrow }X \) implies \( {X}_{n}\overset{\mathrm{r}}{ \rightarrow }X \in {L}_{r} \) . | Observe that for large \( n,{\left. {\int }_{A}{X}_{n}\right| }^{r} \leqq {\int }_{A}{Y}^{r} \). We proved in 9.3 a particular case of this corollary, with \( Y = c < \infty \) . | No |
Corollary 1. If \( {P}_{n}\overset{w}{ \rightarrow }P \) then \( {P}_{n}{h}^{-1}\overset{w}{ \rightarrow }P{h}^{-1} \) for every \( P \) -a.e. continuous \( h \) on \( \mathfrak{X} \) to \( {\mathfrak{X}}^{\prime } \), equivalently \( \int g\left( h\right) d{P}_{n} \rightarrow \int {gd}\left( {{P}_{n}{h}^{-1}}\right) \... | For, \( P{H}_{h} = 0 \) and \( \left( \overline{{h}^{-1}C}\right) \subset \left( {{h}^{-1}C}\right) \cup {D}_{h} \) for every closed \( C \), imply \( P\left( \overline{{h}^{-1}C}\right) = P\left( {{h}^{-1}C}\right) \) hence limsup \( {P}_{n}\left( {{h}^{-1}C}\right) \leqq \) limsup \( {P}_{n}\left( \overline{{h}^{-1}C... | Yes |
Corollary 2. If \( {P}_{n} \rightarrow P \) on \( \mathcal{C} \subset \mathcal{S} \) where \( \mathcal{C} \) is closed under finite intersections and each open set is a countable union of members of \( \mathcal{Q} \), then \( {P}_{n}\xrightarrow[]{w}P \) . | Proof. Let \( U = \mathop{\bigcup }\limits_{{k = 1}}^{\infty }{A}_{k},{A}_{k} \in \mathcal{C} \) . By hypothesis,\n\n\( {P}_{n}\left( {{A}_{1} \cup {A}_{2}}\right) \)\n\n\[= {P}_{n}\left( {A}_{1}\right) + {P}_{n}\left( {A}_{2}\right) - {P}_{n}\left( {{A}_{1}{A}_{2}}\right) \rightarrow P\left( {A}_{1}\right) + P\left( {... | Yes |
Corollary 3. Let \( \mathfrak{X} \) be separable and let \( {P}_{n} \rightarrow P \) on \( \mathcal{C} \subset \mathcal{S} \) . Then \( {P}_{n}\overset{w}{ \rightarrow }P \) if\n\n(i) \( \mathcal{C} \) is closed under finite intersections and, given \( \epsilon > 0 \) and open \( U \) , for every \( x \in U \) there is... | Proof. (i): Since \( \mathfrak{X} \) is separable, given open \( U \), there is a sequence \( \left( {A}_{n}\right) \) in \( \mathcal{C} \) with \( U = \cup {A}_{n}^{ \circ } \) and \( {A}_{n} \subset U \) so that \( U = \cup {A}_{n} \) and Corollary 2 applies.\n\nNote that the second condition on \( \mathcal{C} \) in ... | Yes |
Corollary 1. Every sequence \( {f}_{n} \) of integral ch.f.’s is compact in the sense of ordinary convergence on \( R \) . | For, in view of the above criterion, this statement is equivalent to the weak compactness theorem for d.f.'s. | No |
Corollary 2. If \( {f}_{n} \rightarrow g \) a.e., then \( {F}_{n}\overset{w}{ \rightarrow }F \) up to additive constants, with \( f = {ga}.e \) . | Proof. Since \( {f}_{n} \rightarrow g \) a.e. and the \( {f}_{n} \) are continuous and uniformly bounded by 1, it follows that \( g \) is measurable and bounded a.e. so that, by the dominated convergence theorem, \( {\widehat{f}}_{n} \rightarrow \widehat{g} \) where \( \widehat{g} \) is defined on \( R \) by the Lebesg... | Yes |
Corollary 1. A product of ch.f.'s is a ch.f. and, in particular, iff is a ch.f. so is \( {\left| f\right| }^{2} \) . | For \( f = {f}_{1}{f}_{2} \) is the ch.f. of the d.f. \( F = {F}_{1} * {F}_{2} \), and the particular case follows from the fact that, if \( f \) is a ch.f., so is its complex-conjugate \( \bar{f} \) which corresponds to the d.f. \( F\left( {+\infty }\right) - F\left( {-x + 0}\right) \) . | Yes |
Corollary 2. Composition of d.f.'s is commutative and associative. | For the corresponding multiplication of ch.f.'s has these properties. | No |
If the r.v.’s \( {X}_{n} \) are independent and \( {X}_{n}\overset{\text{ a.s. }}{ \rightarrow }0 \), then \( \sum P\left\lbrack {\left| {X}_{n}\right| \geqq c}\right\rbrack < \infty \) whatever be the finite number \( c > 0 \) . | For \( {X}_{n}\overset{\text{ a.s. }}{ \rightarrow }0 \) implies that, if \( {A}_{n} = \left\lbrack {\left| {X}_{n}\right| \geqq c}\right\rbrack \), then \( P\left( {\limsup {A}_{n}}\right) \) \( = 0 \), and independence of the \( {X}_{n} \) implies that of the \( {A}_{n} \) . | No |
Corollary 1. If \( {X}_{n} - {a}_{n}\overset{\mathrm{P}}{ \rightarrow }0 \), then \( {X}_{n}{}^{s}\overset{\mathrm{P}}{ \rightarrow }0 \) and \( {a}_{n} - \mu {X}_{n} \rightarrow 0 \) , and conversely. | This follows by letting \( n \rightarrow \infty \) in (ii) where \( X \) is replaced by \( {X}_{n} \) . | No |
Corollary 2. For \( r > 0 \) and every \( a \) ,\n\n\[ \n\frac{1}{2}E{\left| X - \mu X\right| }^{r} \leqq E{\left| {X}^{s}\right| }^{r} \leqq 2{c}_{r}E{\left| X - a\right| }^{r} \n\]\n\nwhere \( {c}_{r} = 1 \) or \( {2}^{r - 1} \) according as \( r \leqq 1 \) or \( r \geqq 1 \) . | Proof. The right-hand side inequality follows, by the \( {c}_{r} \) -inequality, from\n\n\[ \nE{\left| {X}^{s}\right| }^{r} = E{\left| \left( X - a\right) - \left( {X}^{\prime } - a\right) \right| }^{r} \leqq {c}_{r}E{\left| X - a\right| }^{r} + {c}_{r}E{\left| {X}^{\prime } - a\right| }^{r} \n\]\n\n\[ \n= 2{c}_{r}E{\l... | Yes |
Corollary 1. A series \( \sum {X}_{n} \) of independent r.v.’s converges a.s. if, and only if, \( \mathop{\prod }\limits_{{k = 1}}^{n}{f}_{k} \rightarrow f \) and \( f \) is continuous at the origin or \( f \neq 0 \) on a set of positive Lebesgue measure. | This follows by the continuity theorem or \( {12.4},{4}^{ \circ } \). | No |
Corollary 2. A series \( \sum {X}_{n} \) of independent r.v.’s is essentially convergent or divergent according as | This follows by \( {13.4},{4}^{ \circ } \) and \( \mathbf{b} \) . | No |
Corollary 1. Under the uan condition, \( \mathop{\max }\limits_{k}\left| {{\bar{f}}_{nk} - 1}\right| \rightarrow 0 \) uniformly on every finite interval. | Since, by \( \mathbf{a},\mathop{\max }\limits_{k}\left| {a}_{nk}\right| \leqq \mathop{\max }\limits_{k}{\int }_{\left| x\right| < \tau }\left| x\right| d{F}_{nk} \rightarrow 0 \), the r.v.’s \( {\bar{X}}_{nk} = \) \( {X}_{nk} - {a}_{nk} \) obey the uan condition, and the assertion follows by \( \mathbf{A} \). | No |
Corollary 2. Under the uan condition, given \( b < \infty \), all \( \log {f}_{nk}\left( u\right) \) exist and are finite for \( \left| u\right| \leqq b \) and \( n \geqq {n}_{b} \) sufficiently large, and\n\n\[ \log {f}_{nk}\left( u\right) = {f}_{nk}\left( u\right) - 1 + {\theta }_{nk}{\left| {f}_{nk}\left( u\right) -... | This follows from \( \mathbf{A} \) and \( \log z = \left( {z - 1}\right) + {\left| \theta \left| z - 1{\left. \right| }^{2}\text{for}\right| z - 1\right| }^{\prime } < \frac{1}{2} \) . | No |
If \( {X}_{k} \) are independent summands and \( {b}_{n} \uparrow \infty \), then \( \mathfrak{L}\left( \frac{{S}_{n}}{{b}_{n}}\right) \rightarrow 0 \) if, and only if, for every \( \epsilon > 0 \) | (i)\n\n\[ \mathop{\sum }\limits_{k}{\int }_{\left| x\right| \geqq \epsilon {b}_{n}}d{F}_{k} \rightarrow 0 \]\n\n\[ \frac{1}{{b}_{n}{}^{2}}\mathop{\sum }\limits_{k}\left\{ {{\int }_{\left| x\right| < {b}_{n}}{x}^{2}d{F}_{k} - {\left( {\int }_{\left| x\right| < {b}_{n}}xd{F}_{k}\right) }^{2}}\right\} \rightarrow 0, \]\n\... | Yes |
If \( {X}_{k} \) are independent summands, then \( \mathfrak{L}\left( \frac{{S}_{n}}{n}\right) \rightarrow \mathfrak{L}\left( 0\right) \) if, and only if,\n\n(i)\n\n\[ \mathop{\sum }\limits_{k}{\int }_{\left| x\right| \geqq n}d{F}_{k} \rightarrow 0 \]\n\n(ii)\n\n\[ \frac{1}{{n}^{2}}\mathop{\sum }\limits_{k}\left\{ {{\i... | This is the classical degenerate convergence criterion. | No |
Corollary 2. If \( {EX} = 0 \) then the random walk is recurrent. | Follows by elementary computations from the fact that, given \( \epsilon > 0 \) , \( {EX} = 0 \) implies \( 0 \leqq 1 - \operatorname{Ref}\left( u\right) < {\epsilon u} \) for \( \left| u\right| < \delta \) sufficiently small. | No |
Corollary 1. If \( {P}^{\mathfrak{B}} \) is a regular c.pr., and one of the \( \sigma \) -fields \( \mathbf{a} \) or \( \mathbf{\pi } \) or \( {\mathcal{B}}_{P} \) is separable, then the foregoing decomposition holds. | Proof. Since atoms of \( \circledast \) and of \( { \circledast }_{P}\left( { \subset \circledast \circledast }\right) \) are \( {P}^{ \circledast } \) -indecomposable, we have only to prove the assertion when \( \alpha \) is separable. Thus, let \( \left\{ {A}_{j}\right\} \) be the countable class which generates \( a... | Yes |
Corollary 2. Let \( \varphi \) on \( {\mathcal{B}}_{0} \) be a \( \sigma \) -finite signed measure. If, given any \( \epsilon > 0 \), for \( k \) sufficiently large\n\n\[ \left| {{\varphi }_{k}\left( B\right) - \varphi \left( B\right) }\right| \leqq {\epsilon PB} \]\n\nwhatever be \( B \in \mathcal{B}\left( {{X}_{k},{X... | Proof. In the first case, hypothesis \( \left( {\mathrm{H}}^{\prime }\right) \) holds, since \( \varphi \) extends to \( \sigma \) and, for \( m \) sufficiently large,\n\n\[ \left| {\mathop{\sum }\limits_{{k = m}}^{n}{\varphi }_{k}\left( {{B}_{kn}C}\right) - \varphi \left( {\mathop{\sum }\limits_{{k = m}}^{n}{B}_{kn}C}... | Yes |
Corollary 2. A translation \( T \) on \( {I}_{a} \) has a unique extension to translation \( T \) on \( \mathfrak{M} \) : a linear continuous transformation with \( {T1} = 1 \) and \( T{I}_{AB} = T{I}_{A} \cdot T{I}_{B} \) | This follows from Corollary 1. | No |
Corollary 2. If \( {X}^{\prime }\left( t\right) \) on \( T \) exists, then \( {\Gamma }_{{X}^{\prime }}\left( {t,{t}^{\prime }}\right) = \frac{{\partial }^{2}}{\partial t\partial {t}^{\prime }}{\Gamma }_{X}\left( {t,{t}^{\prime }}\right) \) on \( T \times T \) . | This property extends at once to\n\n\[ E{X}^{\left( n\right) }\left( t\right) {\bar{X}}^{\left( {n}^{\prime }\right) }\left( {t}^{\prime }\right) = \frac{{\partial }^{n + {n}^{\prime }}}{\partial {t}^{n}\partial {t}^{\prime {n}^{\prime }}}\Gamma \left( {t,{t}^{\prime }}\right) . \] | No |
Corollary 2. If on \( I \times I \) the random function \( X\left( t\right) \) is continuous in q.m. and independent of the increment random function \( {\Delta Y}\left( {t}^{\prime }\right) \), and if the covariance \( {\Gamma }_{X}\left( {t,{t}^{\prime }}\right) \) is bounded while the covariance \( {\Gamma }_{Y}\lef... | Let us observe that, under the bounded variation condition, the covariance \( {\Gamma }_{Y}\left( {t,{t}^{\prime }}\right) \) can be assumed to have the property \( \mathop{\lim }\limits_{{t,{t}^{\prime } \rightarrow - \infty }}{\Gamma }_{Y}\left( {t,{t}^{\prime }}\right) \) \( = 0 \) ; it can also be normalized, that ... | Yes |
Under the hypotheses of the extension theorem, the random function \( \xi \left( t\right) \) is decomposable into two parts with mutually orthogonal increments: | Extend on \( \bar{R} \), let \( \left\{ {t}_{j}\right\} \) be the (countable) discontinuity set of \( F \), and set\n\n\[ \n{\xi }_{d}\left( t\right) = \mathop{\sum }\limits_{{{t}_{j} < t}}\left( {\xi \left( {{t}_{j} + 0}\right) - \xi \left( {{t}_{j} - 0}\right) }\right) ,\;{\xi }_{c}\left( t\right) = \xi \left( t\righ... | No |
Corollary 3. A random function \( X\left( t\right) = \left\{ {{X}_{u}\left( t\right), u \in U}\right\} \) continuous in q.m. on \( R \) is second order stationary if, and only if, | \[ E{X}_{u}\left( t\right) {\bar{X}}_{{u}^{\prime }}\left( {t}^{\prime }\right) = \int {e}^{i\left( {t - {t}^{\prime }}\right) s}d{F}_{u,{u}^{\prime }}\left( s\right) ,\;u,{u}^{\prime } \in U \] where the functions \( {F}_{u,{u}^{\prime }}\left( s\right) \) are of bounded variation on \( R \) and the functions \( {\Del... | Yes |
Corollary 4. A random function \( X\left( z\right), z = u + {it} \), continuous in q.m. in the complex-plane strip \( S : a < u < b \), is second order stationary if, and only if, for \( z,{z}^{\prime } \in S \), \[ {\Gamma }_{X}\left( {z,{z}^{\prime }}\right) = \int {e}^{\left( {z + {\dot{z}}^{\prime }}\right) s}{dF}\... | REMARK. By replacing the argument \( t \) by a complex argument \( z \) in the preceding extension, Corollary 3 extends to random functions of \( z \) . It suffices to replace therein \( t \) by \( z \) and \( {t}^{\prime } \) by \( {\bar{z}}^{\prime } \) . | No |
Corollary 2. If the covariance \( {\Gamma }_{X}\left( {t,{t}^{\prime }}\right) \) is continuous on a closed interval \( I \), then the random function \( X\left( t\right) \) is normal if, and only if, the r.v.’s \( {\lambda }_{n}{\xi }_{n} = {\int }_{I}X\left( t\right) {\psi }_{n}\left( t\right) {dt} \) are normal. | If, moreover, \( X\left( t\right) \) is real-valued, then the \( {\xi }_{n} \) are independent and the proper decomposition of \( X\left( t\right) \) converges a.s. | No |
Corollary 3. If the continuous covariance \( {\Gamma }_{X}\left( {t,{t}^{\prime }}\right) \) is stationary, then (i) The random function \( X\left( t\right) \) is normal if, and only if, the random function \( \xi \left( s\right) \) which figures in its harmonic decomposition \( X\left( t\right) = \int {e}^{its}{d\xi }... | Only the very last assertion deserves proof. It is due to the fact that \( E{\left| d\xi \left( s\right) \right| }^{2} = d{F}_{c}\left( s\right) \), where the function \( {F}_{c}\left( s\right) \) on \( R \) is nondecreasing, bounded, and continuous. Thus, \( R \) can be subdivided into intervals \( I \) on which the i... | No |
Corollary 1. The joint distribution of \( \left( {{M}_{t},{W}_{t}}\right) \) has pr. density\n\n\[ p\left( {x, y}\right) = \sqrt{2/{\pi t}}\frac{{2x} - y}{t}\exp \left\{ {-{\left( 2x - y\right) }^{2}/{2t}}\right\} \]\n\nfor \( x > 0 \) and \( y \leqq x \) and is 0 otherwise. | Upon setting \( y = x - z \) so that \( z \) is a value of \( {M}_{t} - {W}_{t} \), Corollary 1 yields\n\nCorollary 2. The joint | No |
Corollary 2. The joint distribution of \( \left( {{M}_{t},{M}_{t} - {W}_{t}}\right) \) has pr. density\n\n\[ \n{p}_{1}\left( {x, z}\right) = \sqrt{2/{\pi t}}\frac{x + z}{t}\exp \left\{ {{\left( x + z\right) }^{2}/{2t}}\right\} \n\]\n\nfor \( x > 0, z > 0 \) and is 0 otherwise. | Since \( {p}_{1}\left( {x, a}\right) \) is symmetric in \( x \) and \( z \), upon taking c into account, we obtain | No |
Corollary 3. The r.v.’s \( {M}_{t}, - {m}_{t},\left| {W}_{t}\right| ,{M}_{t} - {W}_{t},{W}_{t} - {m}_{t} \) have the same d.f. | \[ F\left( x\right) = \sqrt{2/{\pi t}}{\int }_{0}^{x}{e}^{-{v}^{2}/{2t}}{dv} \] for \( x > 0 \) and 0 otherwise. | Yes |
Theorem 3. Combinatorial equivalence is an equivalence relation. | Proof (SKETCH). By Theorem 1, the composition of two piecewise linear homeomorphisms is a PLH. Now use Theorem 2. | No |
Theorem 4 (Invariance of domain). Let \( U \) be a subset of \( {\mathbf{R}}^{n} \), such that \( U \) is homeomorphic to \( {\mathbf{R}}^{n} \) . Then \( U \) is open. | See W. Hurewicz and H. Wallman [HW], p. 95. | No |
Theorem 1. In a topological space \( \left\lbrack {X,\mathcal{O}}\right\rbrack \), let \( G \) be a collection of pathwise connected sets, with a point \( P \) in common. Then the union \( {G}^{ * } \) of the elements of \( G \) is pathwise connected. | Proof. Given \( Q \in {g}_{Q} \in G, R \in {g}_{R} \in G \), let \( p \) be a path in \( {g}_{Q} \), from \( Q \) to \( P \) , and let \( q \) be a path in \( {g}_{R} \), from \( P \) to \( R \) . Then \( p \) and \( q \) fit together to give a path \( r \), in \( {g}_{O} \cup {g}_{R} \subset {G}^{ * } \), from \( Q \)... | Yes |
Theorem 2. Pathwise connectivity is preserved by surjective mappings. That is, if \( f : M \rightarrow N \) is a mapping, and \( M \) is pathwise connected, then so also is \( N \). | Proof. Given \( P, Q \in N \), take \( {P}^{\prime },{Q}^{\prime } \in M \) such that \( f\left( {P}^{\prime }\right) = P \) and \( f\left( {Q}^{\prime }\right) \) \( = Q \) ; and let \( p \) be a path in \( M \) from \( {P}^{\prime } \) to \( {Q}^{\prime } \) . Then \( f\left( p\right) \) is a path in \( N \) from \( ... | Yes |
Theorem 3. Every simplex is pathwise connected. | Proof. Because it is convex. | No |
Theorem 4. Let \( K \) be a complex. If \( K \) is connected, then \( \left| K\right| \) is pathwise connected. | Proof. Let \( {v}_{0} \in {K}^{0} \) . We shall show that for each \( v \in {K}^{0} \) there is a path in \( \left| {K}^{1}\right| \) from \( {v}_{0} \) to \( v \) . Let \( V \) be the set of all vertices \( v \) of \( K \) that have this property, and let \( {K}_{1} \) be the set of all simplexes of \( K \) all of who... | Yes |
Theorem 5. Given \( M \subset X, M = H \cup K \) . Then (1) \( H \) and \( K \) are separated if and only if \( \left( 2\right) H, K \in \mathcal{O} \mid M \) and \( H \cap K = \varnothing \) . | Proof. Suppose that (1) holds. Let \( U \) be the union of all open sets that intersect \( H \) but not \( K \) . Then \( H \subset U \) and \( U \cap K = \varnothing \), so that \( H = M \cap U \) \( \in \mathcal{O} \mid M \) . Similarly, \( K \in \mathcal{O} \mid M \) . Therefore (2) holds.\n\nSuppose, conversely, th... | Yes |
Theorem 7. For spaces, connectivity is preserved by surjective mappings. That is, if \( \left\lbrack {X,\mathcal{O}}\right\rbrack \) is connected, and \( f : X \rightarrow Y \) is a mapping, then \( \left\lbrack {Y,{\mathcal{O}}^{\prime }}\right\rbrack \) is connected. | Proof. Suppose not. Then \( Y = U \cup V \), where \( U \) and \( V \) are disjoint, open, and nonempty. Therefore \( X = {f}^{-1}\left( U\right) \cup {f}^{-1}\left( V\right) \), and the latter sets are disjoint, open, and nonempty, which is impossible. | Yes |
Theorem 8. For sets, connectivity is preserved by surjective mappings. | Proof. By the preceding two theorems. | No |
Theorem 9. Every closed interval in \( \mathbf{R} \) is connected. | Proof. This turns out to be the \( n \) th formulation of the continuity of \( \mathbf{R} \) . Suppose that \( \left\lbrack {a, b}\right\rbrack = H \cup K \) (separated), with \( a \in H \) . Let\n\n\[ M = \{ x \mid x = a\text{ or }\left\lbrack {a, x}\right\rbrack \subset H\} . \]\n\nThen \( M \) is bounded above. Let ... | Yes |
Theorem 10. If \( H \) and \( K \) are separated, then every connected subset \( M \) of \( H \cup K \) lies either in \( H \) or in \( K \) . | Proof. If not, \( M = \left( {M \cap H}\right) \cup \left( {M \cap K}\right) \), where the two sets on the right are separated and nonempty. (Evidently, if \( H \) and \( K \) are separated, and \( {H}^{\prime } \subset H \) and \( {K}^{\prime } \subset K \), then \( {H}^{\prime } \) and \( {K}^{\prime } \) are separat... | No |
Theorem 11. Every pathwise connected set is connected. | Proof. Suppose that \( M \) is pathwise connected but not connected, so that \( M = H \cup K \) (separated and nonempty). Take \( P \in H, Q \in K \) ; and let \( p \) be a path from \( P \) to \( Q \) in \( M \) . By Theorems 8 and 9, the image \( \left| p\right| = p\left( \left\lbrack {a, b}\right\rbrack \right) \) \... | Yes |
Theorem 12. Let \( K \) be a complex. Then the following conditions are equivalent:\n\n(1) \( K \) is connected.\n\n(2) \( \left| K\right| \) is pathwise connected.\n\n(3) \( \left| K\right| \) is connected. | Proof. (1) \( \Rightarrow \) (2), by Theorem 4. (2) \( \Rightarrow \) (3), by Theorem 11. Suppose, finally, that (1) is false, so that \( K = {K}_{1} \cup {K}_{2} \), where \( {K}_{1} \) and \( {K}_{2} \) are disjoint nonempty complexes. From Condition K. 3 of the definition of a complex,\nit follows that no point \( v... | Yes |
Theorem 13. In \( {\mathbf{R}}^{n} \), every connected open set \( U \) is broken-line-wise connected. | Proof. Let \( P \in U \), and let \( V \) the union of \( \{ P\} \) and the set of all points of \( U \) that can be joined to \( P \) by broken lines lying in \( U \) . It is then easy to show that both \( U \) and \( U - V \) are open. If \( U - V \neq \varnothing \), then \( U \) is the union of two disjoint nonempt... | No |
Theorem 14. Let \( G \) be a collection of connected sets, with a point \( P \) in common. Then the union \( {G}^{ * } \) of the elements of \( G \) is connected. | Proof. Suppose that \( {G}^{ * } = H \cup K \) (separated and nonempty), with \( P \in H \) . Since each \( g \in G \) is connected, each \( g \) lies in \( H \) or in \( K \) . Therefore \( g \subset H \) , \( {G}^{ * } \subset H \), and \( K = \varnothing \), which contradicts the hypothesis for \( K \) . | Yes |
Theorem 15. If \( M \) is connected, and \( M \subset L \subset \bar{M} \), then \( L \) is connected. | Proof. Suppose that \( L = H \cup K \) (separated and nonempty). Let \( {H}^{\prime } = M \) \( \cap H \) and \( {K}^{\prime } = M \cap K \), so that \( M = {H}^{\prime } \cup {K}^{\prime } \) . Then \( {H}^{\prime } \) and \( {K}^{\prime } \) are separated. Now \( H \) contains a point \( P \) of \( L \), and \( P \) ... | Yes |
Theorem 1. Let \( J \) be a polygon in \( {\mathbf{R}}^{2} \) . Then \( {\mathbf{R}}^{2} - J \) has exactly two components. | Proof. Let \( N \) be a \ | No |
Lemma 1. \( {\mathbf{R}}^{2} - J \) has at most two components. | Proof. Starting at any point \( P \) of \( N - J \), we can work our way around the polygon, along a path in \( N - J \), until we get to either \( {P}_{1} \) or \( {P}_{2} \) . (See Figure 2.2.) From this the lemma follows, because every point \( Q \) of \( {\mathbf{R}}^{2} - J \) can be joined to some point \( P \) o... | No |
Lemma 2. \( {\mathbf{R}}^{2} - J \) has at least two components. | Proof. We choose the axes in general position, in the sense that no horizontal line contains more than one of the vertices of \( J \) . (This can be done, because there are only a finite number of directions that we need to avoid. Hereafter, the phrase \ | No |
Theorem 2. Let \( I \) be the interior of the polygon \( J \) in \( {\mathbf{R}}^{2} \) . Then \( \bar{I} \) is a finite polyhedron. That is, there is a finite complex \( K \) in \( {\mathbf{R}}^{2} \) such that \( \left| K\right| = \bar{I} \) . | Proof. Let \( {L}_{1},{L}_{2},\ldots ,{L}_{n} \) be the lines that contain edges of \( J \) . These lines are finite in number, and each intersects the union of the others in a finite number of points. Note that some sets \( {L}_{i} \cap I \) may not be connected; this does not matter. Each line \( {L}_{i} \) decompose... | Yes |
Theorem 3. No broken line separates \( {\mathbf{R}}^{2} \) . That is, if \( B \) is a broken line in \( {\mathbf{R}}^{2} \) , then \( {\mathbf{R}}^{2} - B \) is connected. | Proof. Form a strip-neighborhood \( N \) of \( B \) . As in the proof of Lemma 1 in the proof of Theorem 1, each point \( P \) of \( N - B \) can be joined to either \( {P}_{1} \) or \( {P}_{2} \) by a path in \( N - B \) . (See Figure 2.5.) But if \( {P}_{1} \) and \( {P}_{2} \) are near an end-point, as in the figure... | Yes |
Theorem 4. Let \( X \) be a topological space and let \( U \) be an open set. Then \( \operatorname{Fr}U = \bar{U} - U. \) | Proof. By definition, \( \operatorname{Fr}U = \bar{U} \cap \overline{X - U} \) . Therefore \( \operatorname{Fr}U \subset \bar{U} \) . Since \( U \) is open, we have \( U \cap \overline{X - U} = \varnothing \) . Since \( \operatorname{Fr}U \subset \overline{X - U} \), it follows that \( \operatorname{Fr}U \) \( \subset ... | Yes |
Theorem 5. Let \( J \) be a polygon in \( {\mathbf{R}}^{2} \), with interior \( I \) and exterior \( E \) . Then every point of \( J \) is a limit point both of \( I \) and of \( E \) . | Proof. Let \( F = \operatorname{Fr}I = \bar{I} - I \) . Then \( F \) separates \( {\mathbf{R}}^{2} \) :\n\n\[ \n{\mathbf{R}}^{2} - F = I \cup \left( {{\mathbf{R}}^{2} - \bar{I}}\right) \n\]\n\nand the sets on the right are disjoint, open, and nonempty; \( {\mathbf{R}}^{2} - \bar{I} \) contains \( E;F \subset J \), and ... | No |
Theorem 6. Let \( J, I \), and \( E \) be as in Theorem 5. Then\n\n\[ J = \operatorname{Fr}I = \operatorname{Fr}E. \] | Proof. \( J \subset \bar{I} \), and \( J \cap I = \varnothing \) . Therefore \( J \subset \bar{I} - I = \operatorname{Fr}I \) . And \( \bar{I} - I \subset J \) , because \( E \) is open. Therefore \( J = \operatorname{Fr}I \) . Similarly, \( J = \operatorname{Fr}E \) . | Yes |
Theorem 7. Let \( M = {B}_{1} \cup {B}_{2} \cup {B}_{3} \) be a polyhedral \( \theta \) -graph in \( {\mathbf{R}}^{2} \), with Bd \( {B}_{i} = \{ P, Q\} \). Then (1) Every component of \( {\mathbf{R}}^{2} - M \) has a polygon \( {B}_{i} \cup {B}_{j} \) as its frontier, and (2) Exactly one of the sets \( {B}_{i} \) lies... | Proof. (1) Let \( U \) be a component of \( {\mathbf{R}}^{2} - M \). It is easy to see geometrically that if \( \operatorname{Fr}U \) contains a point of a set \( \operatorname{Int}{B}_{i} \), then \( \operatorname{Fr}U \) contains all of \( \operatorname{Int}{B}_{i} \), and therefore all of \( {B}_{i} \). Consider a s... | Yes |
Theorem 8. Let \( {B}_{1},{B}_{2},{B}_{3} \) be as in Theorem 7, with Int \( {B}_{2} \) in the interior \( {I}_{13} \) of \( {B}_{1} \cup {B}_{3} \) . Then\n\n(1) The components of \( {I}_{13} - \operatorname{Int}{B}_{2} \) are the interiors \( {I}_{12} \) and \( {I}_{23} \) of \( {B}_{1} \cup {B}_{2} \) and \( {B}_{2}... | Proof. Let \( {E}_{13} \) be the exterior of \( {B}_{1} \cup {B}_{3} \) . Then the bounded components of \( {\mathbf{R}}^{2} - M = {\mathbf{R}}^{2} - \cup ,{B}_{r} \) lie in \( {I}_{13} \), and each of them has a polygon in \( \cup ,{B}_{r} \) as its frontier. Again consider a small circular neighborhood of \( P \) . (... | Yes |
Theorem 1. Let \( {\sigma }^{n} = {v}_{0}{v}_{1}\ldots {v}_{n} \) and \( {\tau }^{n} = {w}_{0}{w}_{1}\ldots {w}_{n} \) be simplexes in \( {\mathbf{R}}^{m} \). Then there is a simplicial homeomorphism \[ f : {\sigma }^{n} \leftrightarrow {\tau }^{n} \] \[ f : {v}_{i} \mapsto {w}_{i}. \] | Proof. For each \( v = \sum {\alpha }_{i}{v}_{i}\left( {{\alpha }_{i} \geq 0,\sum {\alpha }_{i} = 1}\right) \), define \( f\left( v\right) = \sum {\alpha }_{i}{w}_{i} \). Then \( f \) is bijective, and \( f \) and \( {f}^{-1} \) are continuous. (For details, see Problems 0.10-0.17 . Since \( f \) and \( {f}^{-1} \) are... | No |
In Theorem 1, if \( m = n \), then there is a homeomorphism \( g : {\mathbf{R}}^{n} \leftrightarrow {\mathbf{R}}^{n} \) such that \( g \mid {\sigma }^{n} \) is a simplicial homeomorphism \( {\sigma }^{n} \leftrightarrow {\tau }^{n} \) . | The mapping \( v \mapsto v - {v}_{0} \) is a homeomorphism \( {\mathbf{R}}^{n} \leftrightarrow {\mathbf{R}}^{n} \), and maps every simplex simplicially onto a simplex. The composition of two such mappings has the same properties. Therefore we may assume, with no loss of generality, that \( {v}_{0} \) is the origin in \... | Yes |
Theorem 3. Let \( J \) be a polygon in \( {\mathbf{R}}^{2} \), let \( I \) be the interior of \( J \), and let \( K \) be a triangulation of \( \bar{I} \). If \( K \) has more than one 2-simplex, then \( K \) has a free 2-simplex. | Proof. The theorem in this weak form is hard to prove. But we can prove, by induction, the stronger assertion that \( K \) has at least two free 2-simplexes. If \( K \) has exactly two 2-simplexes, then this is clear. We may assume, then, that \( K \) has more than two 2-simplexes; and we may assume, as an induction hy... | Yes |
Theorem 4. Let \( J \) be a polygon in \( {\mathbf{R}}^{2} \). Then there is a homeomorphism \( h : {\mathbf{R}}^{2} \leftrightarrow {\mathbf{R}}^{2} \), such that \( h\left( J\right) \) is the frontier of a 2-simplex. | Proof. Let \( I \) be the interior of \( J \), and let \( K \) be a triangulation of \( \bar{I} \). Any free 2-simplex of \( K \) can be removed by a homeomorphism \( h : {\mathbf{R}}^{2} \leftrightarrow {\mathbf{R}}^{2} \). CASE 1. Suppose that \( {v}_{0}{v}_{1}{v}_{2} \) is free, with \( {v}_{0}{v}_{1}{v}_{2} \cap \o... | No |
Theorem 5. Let \( J \) and \( {J}^{\prime } \) be polygons in \( {\mathbf{R}}^{2} \) . Then there is a homeomorphism \( h : {\mathbf{R}}^{2} \leftrightarrow {\mathbf{R}}^{2}, J \leftrightarrow {J}^{\prime } \) . | Proof. By Theorem 4 there are homeomorphisms\n\n\[ \n{f}_{1} : {\mathbf{R}}^{2} \leftrightarrow {\mathbf{R}}^{2},\;J \leftrightarrow \operatorname{Fr}{\sigma }^{2}, \n\]\n\n\[ \n{f}_{2} : {\mathbf{R}}^{2} \leftrightarrow {\mathbf{R}}^{2},\;{J}^{\prime } \leftrightarrow \operatorname{Fr}{\tau }^{2}. \n\]\n\nBy Theorem 2... | Yes |
Theorem 7. Let \( J \) be a polygon in \( {\mathbf{R}}^{2} \), with interior \( I \), and let \( U \) be an open set containing \( \bar{I} \) . Then there is a homeomorphism \( h : {\mathbf{R}}^{2} \leftrightarrow {\mathbf{R}}^{2} \), such that (1) \( h\left( J\right) \) is the frontier of a 2-simplex and (2) \( h \mid... | Proof. In the proof of Theorem 4, we choose our homeomorphisms so that each of them satisfies (2). | No |
Theorem 1. Let \( U \) be an open set in \( {\mathbf{R}}^{n} \), and let \( P, Q \in U \) . If \( P \) and \( Q \) are in different components of \( U \), then \( U \) is the union of two disjoint open sets containing \( P \) and \( Q \) respectively. | Proof. Every component of \( U \) is open, because every set \( N\left( {P,\varepsilon }\right) \) is connected. Therefore every union of components of \( U \) is open. Let \( {C}_{P} \) be the component of \( U \) that contains \( P \) . Then \( U = {C}_{P} \cup \left( {U - {C}_{P}}\right) \), where \( U - {C}_{P} \) ... | Yes |
Theorem 2. Let \( I \) be the interior of a polygon in \( {\mathbf{R}}^{2} \), and let \( P, Q, R \), and \( S \) be points of \( \mathrm{{Fr}}I \), appearing in the stated cyclic order on \( \mathrm{{Fr}}I \). Let \( A \) be an arc from \( P \) to \( R \), lying in \( \bar{I} \), such that \( A \cap \operatorname{Fr}I... | Proof. Let \( {Q}^{\prime } \) and \( {S}^{\prime } \) be points of \( I \), near \( Q \) and \( S \), as in the figure. If \( {Q}^{\prime } \) and \( {S}^{\prime } \) are in the same component of \( I - A \), then there is a broken line from \( {Q}^{\prime } \) to \( {S}^{\prime } \) in \( I - A \). Therefore there is... | No |
Theorem 3. Let \( J \) be a topological 1-sphere in \( {\mathbf{R}}^{2} \). Then \( {\mathbf{R}}^{2} - J \) is not connected. | Proof. Let \( \bar{I} \) be a polyhedral 2-cell containing \( J \), such that \( J \cap \operatorname{Fr}I \) contains exactly two points \( P \) and \( R \). (Fill in the details for the construction of such an \( \bar{I} \).) Then \( J \) is the union of two arcs \( {A}_{1} \) and \( {A}_{2} \), from \( P \) to \( R ... | No |
Lemma 1. \( {A}_{2} \) contains a point of \( B \), following \( X \) in the order from \( S \) to \( Q \) on B. | Proof. Suppose not. Let \( {B}_{1} \) be the arc \( {ST} \) in \( B \) ; let \( {B}_{2} \) be the arc \( {TX} \) in \( {A}_{1} \) ; and let \( {B}_{3} \) be the arc \( {XQ} \) in \( B \) . Then \( {B}_{1} \cup {B}_{2} \cup {B}_{3} \) is an arc \( {SQ} \), in \( \bar{I} - {A}_{2} \) . Therefore \( S \) and \( Q \) lie i... | Yes |
Lemma 2. \( Z \) lies in a bounded component of \( {\mathbf{R}}^{2} - J \) . | Proof. Suppose not. Then there is a broken line \( {B}_{1} \), from \( Z \) to a point \( W \) of \( \operatorname{Fr}I \), with \( {B}_{1} \subset {\mathbf{R}}^{2} - J \) . We may suppose that \( {B}_{1} \cap \operatorname{Fr}I = W \), since otherwise a shorter broken line would have the same properties. Consider firs... | Yes |
Theorem 4. Let \( I, P, Q, R \), and \( S \) be as in the preceding theorems, and let \( {A}_{1} \) and \( {A}_{2} \) be disjoint arcs in \( \bar{I} \), such that \( {A}_{1} \cap \operatorname{Fr}I = \{ P\} \) and \( {A}_{2} \cap \operatorname{Fr}I = \) \( \{ R\} \) . Then \( S \) and \( Q \) are in the frontier of the... | Proof. In Figure 4.5, we have shown \( \bar{I} \) as a rectangular region, with \( P \) and \( R \) as the midpoints of a pair of opposite sides. See Theorem 3.5 and Problem 3.5.\n\nBy a brick-decomposition of the plane we mean a collection \( G = \left\{ {g}_{i}\right\} \) of polyhedral disks ( \( = 2 \) -cells) such ... | Yes |
Theorem 5. No arc separates \( {\mathbf{R}}^{2} \) . | Proof. Let \( A \) be an arc in \( {\mathbf{R}}^{2} \) . Since \( A \) is bounded, \( {\mathbf{R}}^{2} - A \) has exactly one unbounded component. Thus we need to show that \( {\mathbf{R}}^{2} - A \) has no bounded component.\n\nIf \( U \) is a bounded component of \( {\mathbf{R}}^{2} - A \), then \( \operatorname{Fr}U... | Yes |
Theorem 6. Let \( J \) be a 1-sphere in \( {\mathbf{R}}^{2} \), and let \( U \) be a component of \( {\mathbf{R}}^{2} - J \). Then \( J = \mathrm{{Fr}}U \) . | Proof. Obviously \( \operatorname{Fr}U \subset J \). If \( \operatorname{Fr}U \) is not all of \( J \), then \( \operatorname{Fr}U \) lies in an arc \( A \) in \( J \). Since \( {\mathbf{R}}^{2} - J \) has another component \( V \), it follows that \( A \) separates \( {\mathbf{R}}^{2} \), which contradicts Theorem 5. | Yes |
Theorem 7. Let \( J \) be a 1-sphere in \( {\mathbf{R}}^{2} \). Then \( {\mathbf{R}}^{2} - J \) has only one bounded component. | Proof. Let \( X, Y \), and \( Z \) be as in Figure 4.2. Here, as usual, \( J = {A}_{1} \cup {A}_{2};T \) is the first point of \( B \) (in the order from \( S \) ) that lies in \( J, T \in {A}_{1};X \) is the last point of \( B \) in \( {A}_{1};Y \) is the first point after \( X \) in \( B \) that lies in \( {A}_{2};Z ... | Yes |
Theorem 8. Let \( K \) be a complex, such that \( M = \left| K\right| \) is a 2-manifold. Then \( K \) is a combinatorial 2-manifold. That is, every subcomplex \( \mathrm{{St}}v \) is a combinatorial 2-cell. | Proof as follows. | No |
Lemma 1. Every vertex of \( K \) lies in an edge of \( K \) . | Proof. Because \( {\mathbf{R}}^{2} \) has no isolated points. | No |
Lemma 2. Every edge \( {v}_{0}{v}_{1} \) of \( K \) lies in at least one 2-simplex of \( K \) . | Proof. Let \( v \) be an interior point of \( {v}_{0}{v}_{1} \), and suppose that \( {v}_{0}{v}_{1} \) lies in no 2-simplex of \( K \) . Then \( v \) has a neighborhood \( U \) in \( \left| K\right| \), lying in Int \( {v}_{0}{v}_{1} \) and homeomorphic to the plane. This is impossible, because no point separates the p... | Yes |
Lemma 3. Every edge of \( K \) lies in at least two 2-simplexes of \( K \) . | Proof. Suppose that \( e \) lies in only one \( {\sigma }^{2} \in K \) ; and let \( v \) and \( U \) be as in the proof of Lemma 2. Then \( {\sigma }^{2} \) contains a semicircle which lies in \( U \) and separates \( U \), and this is impossible, because no arc separates \( {\mathbf{R}}^{2 \) . | Yes |
Lemma 4. Every edge of \( K \) lies in at most two 2-simplexes of \( K \) . | Proof. Suppose that \( e \) is the intersection of three distinct simplexes, \( e = {\sigma }_{1}^{2} \cap {\sigma }_{2}^{2} \cap {\sigma }_{3}^{2} \), and let \( v \) be an interior point of \( e \) . Let \( U \) be a neighborhood of \( v \) in \( K \), homeomorphic to \( {\mathbf{R}}^{2} \), and let \( \varepsilon > ... | Yes |
Lemma 5. Each set \( \left| {L\left( v\right) }\right| \) is connected. | Proof. If not, \( v \) separates \( \left| {\mathrm{{St}}v}\right| \) ; and this is impossible, because no point separates any open set in \( {\mathbf{R}}^{2} \) . | No |
Theorem 9. Let \( K \) be a complex, such that \( M = \left| K\right| \) is a 2-manifold with boundary. Then \( K \) is a combinatorial 2-manifold with boundary, and \( \mathrm{{Bd}}M \) is the union of the edges of \( K \) that lie in only one 2-simplex of \( K \) . | Proof. Let \( e \) be an edge of \( K \) . As in the proof of the preceding theorem, we show that the number of 2-simplexes of \( K \) that contain \( e \) is either 1 or 2 . It follows that every component of every set \( \left| {L\left( v\right) }\right| \) is either a broken line or a 1-sphere. As before, \( \left| ... | Yes |
Theorem 10. Let \( M \) be a 2-manifold with boundary, lying in \( {\mathbf{R}}^{2} \) . If \( M \) is closed, then \( \mathrm{{Bd}}M = \mathrm{{Fr}}M \) . (In the case \( M = \left| K\right| \), it then follows that Fr \( M = \left| {\partial K}\right| \) .) | Proof.\n\n(1) Since \( \operatorname{Fr}M \) is closed, every point of \( M - \operatorname{Fr}M \) has a locally Euclidean open neighborhood in \( M \) . Thus \( M - \operatorname{Fr}M \subset \operatorname{Int}M = M - \) Bd \( M \), and Bd \( M \subset \operatorname{Fr}M \) .\n\n(2) Since \( M \) is closed, \( \opera... | Yes |
Theorem 1. Given \( {K}_{1} < K \) . Then (1) \( f \) is PL relative to \( K \) and \( L \) if and only if (2) \( f \) is PL relative to \( {K}_{1} \) and \( L \) . | Proof. The implication (2) \( \Rightarrow \) (1) is trivial, since any subdivision of \( {K}_{1} \) is a subdivision of \( K \) . It remains to show that \( \left( 1\right) \Rightarrow \left( 2\right) \) .\n\nGiven \( {K}_{2} < K \), such that for each \( \sigma \in {K}_{2}, f \mid \sigma \) is linear. Let \( {K}_{12} ... | Yes |
Theorem 2. Let \( {L}_{1} \) be a subdivision of \( L \) . A homeomorphism \( f : \left| K\right| \leftrightarrow \left| L\right| \) is PL relative to \( K \) and \( L \) if and only if \( f \) is PL relative to \( K \) and \( {L}_{1} \) . | Proof. Here \ | No |
Theorem 4. Let \( {K}_{1} \) and \( {K}_{2} \) be combinatorial 2-cells and let \( f \) be a PLH Bd \( \left| {K}_{1}\right| \leftrightarrow \) Bd \( \left| {K}_{2}\right| \) . Then \( f \) has a PLH extension \( {f}^{\prime } : \left| {K}_{1}\right| \leftrightarrow \left| {K}_{2}\right| \) . | Proof. For \( i = 1,2 \), let \( {g}_{i} \) be a PLH \( {\sigma }^{2} \leftrightarrow \left| {K}_{i}\right| \). Then \( {g}_{2}^{-1}f{g}_{1} \) is a PLH Bd \( {\sigma }^{2} \leftrightarrow \) Bd \( {\sigma }^{2} \). This has a PLH extension \( {g}^{\prime } : {\sigma }^{2} \leftrightarrow {\sigma }^{2} \). Let \( {f}^{... | Yes |
Theorem 5. Let \( K \) be a complex, such that \( {M}^{2} = \left| K\right| \) is a 2-manifold with boundary. Let \( J \) be a polygon in \( \left| K\right| \), that is, a 1-sphere which forms a subcomplex of a subdivision. If \( J \) lies in a set \( \left| {\mathrm{{St}}v}\right| \), then \( J \) is the boundary of a... | Proof. Let \( f \) be a PLH: \( \left| {\operatorname{St}v}\right| \rightarrow {\mathbf{R}}^{2} \) . Then \( f\left( J\right) \) is a polygon in \( {\mathbf{R}}^{2} \), and the interior of \( f\left( J\right) \) in \( {\mathbf{R}}^{2} \) lies in \( f\left( \left| {\mathrm{{St}}v}\right| \right) \) . Let \( I \) be the ... | Yes |
Theorem 6. Let \( {C}_{1} \) and \( {C}_{2} \) be 2-cells, and let \( f \) be a homeomorphism Bd \( {C}_{1} \leftrightarrow \) Bd \( {C}_{2} \) . Then \( f \) has a homeomorphic extension \( {f}^{\prime } : {C}_{1} \leftrightarrow {C}_{2} \) . | Proof. For \( i = 1,2 \), let \( {g}_{i} \) be a homeomorphism \( {\sigma }^{2} \leftrightarrow {C}_{i} \) . Now proceed exactly as in the proof of Theorem 4. | Yes |
Theorem 1. Let \( v{v}^{\prime } \) be a 1-simplex, let \( h \) be a homeomorphism \( v{v}^{\prime } \leftrightarrow A \subset {\mathbf{R}}^{2} \) , with \( v \mapsto P,{v}^{\prime } \mapsto Q \), and let \( \varepsilon \) be a positive number. Then there is a broken line \( B \), from \( P \) to \( Q \), lying in \( N... | Proof. We recall that \( N\left( {A,\varepsilon }\right) = \{ Q \mid d\left( {P, Q}\right) < \varepsilon \) for some \( P \in A\} \) . Obviously \( N\left( {A,\varepsilon }\right) \) is open, and it is easy to show that \( N\left( {A,\varepsilon }\right) \) is connected. (Theorems 1.3 and 1.7, and Problem 1.26.) Theref... | No |
Theorem 2. Let \( {K}^{1} \) be a 1-dimensional complex, not necessarily finite, let \( h \) be a homeomorphism \( \left| {K}^{1}\right| \rightarrow {\mathbf{R}}^{2} \), and let \( \phi \) be \( \gg 0 \) on \( {K}^{1} \) . Then there is a PLH \( f : \left| {K}^{1}\right| \rightarrow {\mathbf{R}}^{2} \), such that (1) \... | Proof. For each edge \( e \) of \( {K}^{1} \), let \( {\varepsilon }_{e} \) be the greatest lower bound inf \( \left( {\phi \mid e}\right) \) of \( \phi \mid e \) . Then \( {\varepsilon }_{e} > 0 \) for each \( e \) . For each \( e \in {K}^{1}, h \mid e \) is uniformly continuous. It follows that there is a subdivision... | Yes |
Theorem 4. Let \( {K}_{1} \) be a combinatorial 2-manifold with boundary, let \( {K}_{2} \) be a combinatorial 2-manifold, let \( h \) be a homeomorphism \( \left| {K}_{1}\right| \rightarrow \left| {K}_{2}\right| \), and let \( \phi \) be \( \gg 0 \) on \( {K}_{1} \) . Then there is a PLH \( f : \left| {K}_{1}\right| \... | Proof. First we observe that Theorem 2 still holds when \( {\mathbf{R}}^{2} \) is replaced by an arbitrary combinatorial 2-manifold \( {K}_{2} \) . The reason is that in \( {K}_{2} \), each complex St \( v \) is a combinatorial 2-cell, and so there is a PLH \( g : \left| {\operatorname{St}v}\right| \leftrightarrow C \)... | Yes |
Lemma 1. In \( {\mathbf{R}}^{{2n} + 1} \), let \( H \) be a hyperplane of dimension \( \leq {2n} \). Then \( H \) contains no open set. | Proof. Because no subspace of dimension \( \leq {2n} \) contains an open set. | No |
Lemma 2. In \( {\mathbf{R}}^{{2n} + 1} \), let \( V = \left\{ {{v}_{0},{v}_{1},\ldots ,{v}_{k}}\right\} \) be a set of \( k + 1 \) points, in general position. Then \( V \) lies in exactly one \( k \) -dimensional hyperplane. | Proof. Because the points \( {v}_{i}^{\prime } = {v}_{i} - {v}_{0}\left( {1 \leq i \leq k}\right) \), being linearly independent, lie in exactly one \( k \) -dimensional subspace. | Yes |
Lemma 3. There is a countable set\n\n\\[ \nV = \\left\\{ {{v}_{1},{v}_{2},\\ldots }\\right\\} \n\\]\n\nof points of \\( {\\mathbf{R}}^{{2n} + 1} \\), with \\( {v}_{i} \\neq {v}_{j} \\) for \\( i \\neq j \\), such that (1) \\( V \\) is in general position and (2) \\( V \\) has no limit point. | Proof. For each \\( v \\in {\\mathbf{R}}^{{2n} + 1} \\), let \\( {x}_{1}\\left( v\\right) \\) be the first coordinate of \\( v \\) . Let \\( {v}_{1} \\) be any point such that \\( {x}_{1}\\left( {v}_{1}\\right) \\geq 1 \\) .\n\nSuppose now that we have given \\( {v}_{1},{v}_{2},\\ldots ,{v}_{m} \\), in general position... | Yes |
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