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\begin{center}
MATH 770 HOMEWORK 4 \\
{\it Semester I, 2006-2007} \\
{\it Due November 2}
\end{center} 


\bigskip

\textbf{16.} Prove that there are only $\cccc$ Borel subsets of $\RRR$.
\emph{Hint.}  See \#15; prove that $|\Sigma_\alpha| = |\Pi_\alpha| = \cccc$
by induction on $\alpha$.

\bigskip

\textbf{17.}
Every countable total order is embeddable into $\QQQ$.
You can do the embedding and preserve some topological properties 
as follows:
Let $A$ be a countably infinite totally ordered set with first and last element.
List $A$ as $\{a_n : n\in\omega\}$,
with $a_0$ is the first element of $A$ and $a_1$
the last element of $A$.
Define $f: A \to [0,1] \cap \QQQ$ so that $f(a_0) = 0$, $f(a_1) = 1$, and
$f(a_n) = (f(a_i) + f(a_j))/2$ when $n \ge 2$, where
$a_i$ is the largest element in $\{a_\ell : \ell < n \ \&\ a_\ell < a_n\}$, and
$a_j$ is the smallest element in $\{a_\ell : \ell < n \ \&\ a_\ell > a_n\}$.
Note that $f$ is an order isomorphism onto $f(A)$.
\begin{itemizz}
\item[1.] If $S \subseteq A$ and \emph{if} $\sup(S)$ exists in $A$,
then $f(\sup(S)) = \sup(f(S))$.
\item[2.] If $S \subseteq A$ and \emph{if} $\inf(S)$ exists in $A$,
then $f(\inf(S)) = \inf(f(S))$.
\item[3.] If $A$ is well-ordered then $f(A)$ compact as a set of reals.
\end{itemizz}
Actually (3) works for any countable compact LOTS.  Also,
since $f(A)$ is obviously bounded, it's sufficient to show that
it's closed in $\RRR$.


\bigskip

\textbf{18.} Let $V$ be an uncountable vector space over $\QQQ$.
Prove that $\dim(V) = |V|$.
Conclude that $\RRR$ and $\CCC$ are isomorphic as abelian groups.
\emph{Hint.} Count linear combinations to show that
$\dim(V) < |V|$ is contradictory.

\bigskip

\textbf{19.} Let $p$ be prime.
Prove that any two abelian groups of exponent $p$
of the same size are isomorphic.
A group has \emph{exponent} $p$ iff $x^p = 1$ for all $x$.
For this problem, it's better to write the groups additively,
and view them as vector spaces over the field $\ZZZ_p = \ZZZ / p\ZZZ$.
\emph{Remark.} It then follows,
by the \L o\'s -- Vaught test in model theory, that the theory of infinite
abelian groups of exponent $p$ is decidable.

\bigskip

\textbf{20.} Prove that it's consistent with ZF that 
$\RRR$ and $\CCC$ are not isomorphic groups.
\emph{Remark and Hint.} A function $f : \RRR \to \RRR$
is called \emph{additive} iff $f(x+y) = f(x) + f(y)$ for all $x,y$.
You may use the fact that it's consistent that all additive
functions are of the form $f(x) = cx$.  This follows because
all Lebesgue measurable additive functions are of this form,
and Solovay (1970) showed that it's consistent that all
sets of reals are measurable.




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