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\centerline{Additions to Part IV: Notes}
\smallskip
\item{I.} Additions to note 120:
\smallskip
Replace 42. by:
\item{42.} $14 + 43 \leftrightarrow 407$, \ac{Bacsich} \cite{1972b}.
and add:
\item{64.} $8 + 70 \leftrightarrow 8 + 406$, \ac{Alas} \cite{1994}.
\item{65.} $8 + 385 \leftrightarrow 8 + 406$, \ac{Alas} \cite{1994}.
\smallskip
\item{II.} Replace note 127 by the following:
\smallskip
\head{NOTE 127}\endhead Forms [1 BZ] (Vector space multiple
choice) and [1 DG] (Vector space Kinna-Wagner principle) were suggested by
K. Keremedis. It is clear that [1 BZ] implies 346.
In this note we prove that [1 DG] implies the
Kinna-Wagner principle $KW(\infty,< \aleph_0)$ (form [62 E]).
Since the axiom of choice is implied by the conjunction of forms
62 and 67, we obtain a proof that [1 DG] + 67 implies the axiom of choice.
(Form 62 is $C(\infty,<\aleph_0)$ and 67 is the axiom of multiple
choice.) Keremedis \cite{1999d} proves that [1 DG] implies 67 to complete
the proof that [1 DG] implies the axiom of choice. Similarly,
since [1 BZ] implies form 67, we obtain the result: [1 BZ] implies
the axiom of choice.
\par
Let $X = \{ y_i : i\in K\}$ be a family of finite sets. For each
$y_i$ let $U_i$ be the real vector space ${\Bbb R}^{y_i}$ with pointwise
addition and scalar multiplication. (If $y_i = \{a_1,\ldots,a_n \}$
we could think of $U_i$ as being the set of all formal sums
$k_1 a_1 + \cdots + k_n a_n$ where the $k_i$'s are real.) Let
$S_i$ be the subspace $\{g\in U_i : g\hbox{ is constant}\}$.
(Or in terms of formal sums, $S_i$ is all formal sums
$k a_1 + \cdots + k a_n$.) Let $V_i$ be the quotient space
$V_i = U_i/S_i$. That is, $V_i$ consists of all equivalence classes
$[g]$ of elements of $U_i$ under the relation $g\sim f
\Leftrightarrow g-f\in S_i$. By form [1 DG] there is a family
$\{ F_i : i\in K\}$ such that for each $i\in K$, $F_i$ is an
independent subset of $V_i$. Since $U_i$ is finite dimensional
$F_i$ must be finite. Say $F_i = \{b_1,\ldots,b_r\}$
then since $F_i$ is independent, the element $w_i = b_1 + \cdots + b_r$
is not zero. The vector $w_i = [g]$ for some $g\in U_i$. Assume
$f\in [g]$. Then if $a,a'\in y_i$ and $g(a)\le g(a')$ it follows
from the fact that $f - g$ is constant that $f(a) \le f(a')$. Therefore
the set $K_i = \{ a\in y_i : g(a)$ is minimum among the numbers
$g(a')$ for $a'\in y_i \}$ is independent of the choice of $g\in w_i$.
It is also true that $K_i \ne y_i$. This follows from the fact
that $w_i \ne 0$ which implies that $g$ is not constant.
The family $\{K_i : i\in K \}$ is therefore a Kinna-Wagner function
for $X$.
\smallskip
\item{III.} Add a new note 150.
\smallskip
\head{NOTE 150}\endhead We give a proof that 385 (Every proper
filter with a countable base over a set $S$ (in ${\Cal P}(S)$) can
be extended to an ultrafilter.) implies 70 (There is a non-trivial
ultrafilter on $\omega$.) It is sufficient to show that there is a
filter on $\omega$ with a countable base. For each $n\in\omega$
let $x_n = \{m\in\omega: m\geq n\}$. Then, $X = \{x_n: n\in\omega\}$
is a countable set which is a base for a filter $\Cal F$ on $\omega$.
\smallskip
\item{IV.} Add the following at the end of note 23:
\smallskip
(A {\it topological vector space (linear topological space)} is a
vector space (linear space) with a topology in which the
operations of addition and scalar multiplication are continuous.
An {\it affine subspace}, $A$, of a topological vector space $E$ is a
translation of a subspace of $E$, $A =v+S = \{v + w: w\in S\}$, where
$S$ is a subspace of $E$ and $v\in E$.)
\smallskip
\item{V.} Add the following right after the definition of
locally convex in note 96:
\smallskip
(A subset $S$ of $X$ is {\it convex} if for all $x$ and $y$ in $S$,
$L(x,y)\subseteq S$, where $L(x,y)= \{\lambda_1 x + \lambda_2 y:
\lambda_1, \lambda_2\in \Bbb R, \lambda_1 + \lambda_2 = 1,
\hbox{ and, } 0\le \lambda_1,\lambda_2\le 1\}$.
\bye