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- (Kunen IV.7.10, part 1) Let
$\aleph_0\leq\kappa\lt\lambda$ and let$\mathbb P=Fn(\kappa,\lambda)$ . Show that$\lambda$ is countable in$V[G]$ , and all cardinals of$V$ above$\lambda$ remain cardinals in$V[G]$ . - (Kunen IV.7.10, part 2) With
$\mathbb P$ as above, show that if$V$ satisfies the GCH then$V[G]$ satisfies the GCH. - (21.1) Show that if
$\mathbb P=Fn(\omega,\omega)$ and$G$ is$V$ -generic, then$g=\bigcup G$ is not dominated by any element of the Baire space of$V$ . - (23.1) Show that random forcing, together with
$[\emptyset]$ , forms a complete Boolean algebra. (A Boolean algebra$B$ is complete if every subset$X\subset B$ has a least upper bound.)
- (19.1) Let
$\mathcal G\subset Fn(\kappa,\omega)$ be uncountable. Show that$\mathcal F=\set{dom(f)\mid f\in\mathcal G}$ is uncountable. - (Kunen III.2.7) If
$\kappa$ is a singular cardinal, show that there is a family$\mathcal F$ of$2$ -element subsets of$\kappa$ with no subfamily of size$\kappa$ that forms a Delta system. Hint: Let$\theta$ be the cofinality of$\kappa$ . Elements of$\mathcal F$ will contain one element below$\theta$ and one element above$\theta$ . - (Kunen IV.3.18) Let
$\mathbb P$ be a countable partial order and let$J$ be a set of size$\aleph_1$ in$V$ . Let$G$ be generic, and let$E$ be an uncountable subset of$J$ in$V[G]$ . Prove that there is an uncountable subset$E'\subset E$ in$V$ . Hint:$E'=\set{j\in J\mid p\Vdash \check j\in\dot E}$ .
- (16.1) In many cases it is more convenient to use
$g=\bigcup G$ instead of$G$ itself. Write down a name$\gamma$ such that$\gamma_G=g$ whenever$G$ is a filter and$g=\bigcup G$ . - (Kunen IV.2.8) Let
$\tau=\set{(\emptyset,p),(\set{(\emptyset,q)},r)}$ . There are eight possibilities for whether$p,q,r$ are$\in G$ or$\notin G$ . Compute$\tau_G$ in all eight cases. - (Kunen IV.2.16) Given
$a\in V[G]$ we showed how to produce a name for$b\in V[G]$ such that$\bigcup a\subset b$ . Show how to produce a name for$b\in V[G]$ such that$\bigcup a=b$ . Hint: Let$\tau$ be a name for$a$ and define the name$\pi=\set{(\theta,p):(\exists(\sigma,q)\in\tau)(\exists r);(\theta,r)\in\sigma\wedge p\leq r\wedge p\leq q}$ . - (Kunen IV.2.28) Give an example of
$\mathbb P$ , a sentence$\psi$ of the forcing language, and distinct generic filters$G,H$ such that$V[G]=V[H]$ and$V[G]\models\psi$ and$V[H]\models\neg\psi$ . Hint: Use a finite$\mathbb P$ so that$V[G]=V[H]=V$ , and write a sentence about the canonical name$\Gamma$ for the generic filter.
- (Jech 14.5) Show that a filter
$G\subset\mathbb P$ meets every dense set of$\mathbb P$ (in$V$ ) if and only if$G$ meets every maximal antichain of$\mathbb P$ (in$V$ ). - (15.1) Assume MA. Let
$\mathcal F$ be an \emph{almost disjoint} family of infinite subsets of$\omega$ : for all$A,A'\in\mathcal F$ we have that$A\cap A'$ is finite. Show that if$\abs{\mathcal F}<\mathfrak c$ then there exists a single infinite set$B$ such that$A\cap B$ is finite for all$A\in\mathcal F$ . [Hint: consider the forcing$\mathbb P$ consisting of pairs$(s,F)$ where$s$ is a finite subset of$\omega$ ,$F$ is a finite subset of$\mathcal F$ , and$(s',F')\leq(s,F)$ iff$s'\supset s$ ,$F'\supset F$ , and whenever$A\in F$ we have$A\cap s'\subset s$ .]
- (Jech 14.1) Show that in the definition of generic filter, we can replace "for all
$p,q\in G$ there exists$r\in G$ with$r\leq p,q$ " with "for all$p,q\in G$ there exists$r$ with$r\leq p,q$ ". [Hint: show the set$\set{r\mid (r\leq p,q)\vee (r\text{ is incompatible with }p)\vee (r\text{ is incompatible with }q)}$ is dense.] - (Jech 14.3) Show that a filter
$G\subset\mathbb P$ meets every dense set of$\mathbb P$ (in$V$ ) if and only if$G$ meets every dense open set of$\mathbb P$ (in$V$ ). - Suppose that
$\mathbb P$ is not atomless. Show that there is a generic filter$G\subset\mathbb P$ . (Recall$\mathbb P$ is atomless if for every$p\in\mathbb P$ there exist$q,r\leq p$ such that$q,r$ are incompatible.) - Suppose that
$\mathbb P$ is atomless. Show that$\mathbb P$ has an infinite antichain.
- (11.1) Let
$P=(I_n)$ and$Q=(J_n)$ be interval partitions of$\omega$ , and let$x,y\in2^\omega$ . Prove that$\mathrm{Diff}(P,x)\subset \mathrm{Diff}(Q,y)$ if and only if for all but finitely many$m$ there exists$n$ such that$I_n\subset J_m$ and$x\restriction I_n=y\restriction I_n$ . - (12.1) If
$A\subset 2^\omega$ is null, then there exists a closed set$K\subset 2^\omega\setminus A$ with the property that whenever$V_s\cap K\neq\emptyset$ we have$V_s\cap K$ is nonnull.
Hint: First show that there exists a closed set$C\subset 2^\omega\setminus A$ which is non-null. Let$D$ be the union of all basic open sets$V_s$ such that$m(V_s\cap C)=0$ , and show that$m(D)=0$ . Finally let$K=C\setminus D$ and show that$K$ has the desired properties.]
- (9.1) Show that there is a homeomorphism between co-countable subsets of
$\mathbb R$ and$2^\omega$ . - (9.2) For
$V_s$ a basic open set of$2^\omega$ , let$m(V_s)=2^{-\abs{s}}$ . Then$m$ extends to a measure on the Borel sets of$2^\omega$ (take this for granted). Show that there is a measure-preserving bijection between$[0,1]$ and$2^\omega$ , after possibly throwing away countable subsets of each. - (10.1) Find a morphism behind the proof of the inequality
$non(\mathcal I)\leq cof(\mathcal I)$ (Lemma 8.5). Check that it is dual to a morphism behind the inequality$add(\mathcal I)\leq cov(\mathcal I)$ . - (10.2) Find a morphism behind the proof of the inequality
$\mathfrak b\leq non(\mathcal M)$ (Theorem 9.4). Check that it is dual to a morphism behind the inequality$cov(\mathcal M)\leq\mathfrak d$ .
- (7.2) Find an example of a meager subset of
$\omega^\omega$ which is not in$\mathcal K_\sigma$ . - (7.3) Let
$X$ be the space$\mathbb R^\omega$ with the product topology. Decide whether$X$ is$\sigma$ -compact. - (8.1) Show that
$\mathrm{cof}(\mathcal K_\sigma)=\mathfrak d$ . - (8.2) Let
$\mathcal I$ be the ideal of countable subsets of$\mathbb R$ . Find the values of the four cardinal characteristics of$\mathcal I$ .
- (5.2) Prove the well-ordering principle: If
$A$ is any set, then there exists a binary relation$\leq$ which is a well-order of$A$ . [Use Zorn's Lemma!] - (5.3) If
$A$ is an infinite set, show that$\abs{A}$ is equal to$\aleph_\alpha$ for some ordinal$\alpha$ . - (6.2) Give an example of an open subset of
$\omega^\omega$ which is not closed. - (6.3) Show that
$\omega^\omega$ is homeomorphic to its product with itself$\omega^\omega\times\omega^\omega$ .
- (3.2) Prove that the properties (a)--(c) of a measure imply continuity from below: if
$A_n$ is an increasing sequence of sets and$A=\bigcup A_n$ , then$m(A)=\sup m(A_n)$ . Then prove continuity from above: if$A_n$ is a decreasing sequence of sets,$m(A_n)$ is finite, and$A=\bigcap A_n$ , then$m(A)=\inf m(A_n)$ . - (3.4) Prove directly from the definition of null set that the null sets are closed under countable unions. (The definition of
$A$ is null: for all$\epsilon>0$ there exist intervals$I_n$ such that$A\subset\bigcup I_n$ and$\sum l(I_n)\lt\epsilon$ .) - (4.1) Show that the following sets are all in bijection with one another:
$\mathbb R$ ,$(0,1)$ ,$(0,\infty)$ ,$\mathcal P(\mathbb N)$ , and$\set{A\in P(\mathbb N)\mid A\text{ is infinite}}$ . - (4.2) Which of the following categories satisfy the analog of the Cantor--Schroder--Bernstein theorem? (That is, monomorphisms
$A\to B\to A$ implies isomorphism$A\cong B$ .) linear orders with order-preserving maps; groups with group homomorphisms; topological spaces with continuous maps; topological spaces with piecewise continuous maps.
- (1.1) With the definition of
$C+C'$ for Dedekind cuts, show that addition is commutative and associative. - (1.4) Show that any two complete ordered fields are isomorphic as ordered fields. [Hint: observe that both must contain a copy of
$\mathbb Q$ which is dense.] - (2.1) Compute the sum of the lengths of all of the intervals removed from
$[0,1]$ in the construction of the Cantor set. What if some fraction other than$1/3$ is removed at each stage? - (2.2) Prove proposition 2.4 in the notes:
$A$ is nowhere dense iff$\bar A$ contains no intervals of positive-length iff$A$ is non-dense in every open set.