## 2002 Changes and additions for the Consequences of the Axiom of Choice Project

1. Changes in January, 2002
Luxemburg, W.A.J./ V\"ath, M.
[2001] The existence of non-trivial bounded functionals implies the Hahn-Banach extension theoreme, J. Anal. and its Applications 20, 267-279. 367-382.
• Add the following to the list of forms
[52 T] On every non-trivial Banach space there is a non-trivial bounded linear functional. Luxemburg/Vath [2001]
FORM 417. On every non-trivial Banach space there is a non-trivial linear functional (bounded or unbounded). Luxemburg/Vath [2001].
• Add the following line to the references for book1
43 417 3 Luxemberg/Vath [2001] (M5($\aleph$))
2. Changes in April-May, 2002
De la Cruz, O.
[2001] Finiteness and choice, Accepted Fund. Math.
Herrlich, H. Keremedis, K. Tachtsis, E.
[2002] Striking Differences between ZF and ZF + Weak Choice in view of Metric Spaces Accepted Quaestiones Math.

Keremedis, K. Tachtsis, E.
[1999b] On the extensibility of closed filters in $T_1$ spaces and the existence of well orderable filter bases, Comment. Mat. Univ. Carolinae Vol. 40, 1-11.
[2000] On Loeb and weakly Loeb Hausdorff spaces, Math. Logic Quart. Vol 46, 35-44.
[2000a] On Lindel\"of metric spaces and weak forms of the axiom of choice, Math. Log. Quart. Vol 46, 35-44.
[2001] Compact metric spaces and weak forms of the axiom of choice, Math. Logic Quart. Vol 47, 117-128.
(Note [1999a] was changed to [2001])

• Add the following lines to the references for book1
341 10 (1) Herrlich/Keremedis/Tachtsis [2002]
383 232 (1) note 158
133 340 (1) note 157
340 341 (1) clear

(Note, since 383 implies 232, it follows that 232 and 383 are equivalent in ZF$^0$ because it was shown in Howard/Keremedis/Rubin/Stanley [2002a] that 232 implies 383. Change forms 383, [383 A]-[383 C] to [232 H]-[232 K].)

3. Make the following changes and additions to the notes

Revision to note 94:

4. Changes in June-July 2002
Tachtsis, E.
[2002] Disasters in metric topology without choice, Comment. Math. Univ. Carolinae, Vol 43, 165-174.

De la Cruz, O., Hall, E., Howard, P., Keremedis, K., Rubin, J. E.
[2001] Products of compact spaces and the axiom of choice, Accepted Math. Logic Quart.
[2002] Products of compact spaces and the axiom of choice II, Accepted Math. Logic Quart.
[2002a] Metric spaces and the axiom of choice , In preparation.

De la Cruz, O., Hall, E., Howard, P., Rubin, J. E., Stanley, A.,
[2002] Definitions of compactness and the axiom of choice, J. of Symbolic Logic Vol 67, 143-161.

• Additions to the Numerical List of Forms:
• Form 18 was the same as [18 B] so replace [18 B] by:
• [18 B] PUT($\aleph_0$,2,WO): The union of a denumerable family of pairs has an infinite well ordered subset. (18 $\to$ [18 B] $\to$ [18 A])

• New Forms:
• [8 AO] $PUT(\aleph_0,\infty,WO)$: The union of a denumerable set of pairwise disjoint non-empty sets has an infinite well ordered subset. (8 $\to$ [8 AO] $\to$ [8 T])

• [10 S] $UT(\aleph_{0},<\aleph_{0},WO)$: The union of denumerably many pairwise disjoint finite sets can be well ordered. ([10 A] $\to$ [10 S] $to$ 10)

• [10 T] $PUT(\aleph_0,<\aleph_0,WO)$: The union of a denumerable set of pairwise disjoint finite sets has an infinite well ordered subset. ([10 A] $\to$ [10 T] $\to$ [10 M])

• [80 B] $UT(\aleph_0,2, WO)$: The union of denumerably many pairs can be well ordered.

• [373 D($n$)] $PUT(\aleph_0,n,WO)$ for $n\in\omega -\{0,1\}$: The union of a denumerable set of pairwise disjoint $n$-element sets has an infinite well ordered subset

• [374 D($n$)] $UT(\aleph_0,n,WO)$, $n\in\omega-\{0,1\}$: The union of a denumerable set of $n$-element sets can be well ordered.

• FORM 421. $UT(\aleph_0,WO,WO)$: The union of a denumerable set of well orderable sets can be well ordered.

• FORM 422($n$). $UT(WO,n,WO)$, $n\in \omega-\{0,1\}$: The union of a well ordered set of $n$ element sets can be well ordered.

• [422 A($n$)] For each $i$, $2\leq i\leq n\in \omega-\{0,1\}$, $C(WO,i)$: Every well ordered family of $i$-element sets has a choice function.

• FORM 423. $C(\aleph_0, n)$ for $n\in\omega-\{0,1\}$: Every denumerable set of $n$ element sets has a choice function.

• Add the following lines to the references for book1
422 47 (1) clear
422 111 (1) clear
343 154 (1) De la Cruz/Hall/Howard/Keremedis/Rubin [2002]
8 173 (3) De la Cruz/Hall/Howard/Keremedis/Rubin [2002a] (N57T)
43 173 (3) De la Cruz/Hall/Howard/Keremedis/Rubin [2002a] (N57T)
421 338 (1) clear
423 373 (1) clear
8 382 (3) De la Cruz/Hall/Howard/Keremedis/Rubin [2002a] (N57T)
43 382 (3) De la Cruz/Hall/Howard/Keremedis/Rubin [2002a] (N57T)
231 421 (1) clear
231 422 (1) clear
10 423 (1) clear
47 423 (1) clear
374 423 (1) clear
10 424 (1) clear

Replace
9 341 (3) De la Cruz/Hall/Howard/Keremedis/Rubin [2002a] (N41T)
by
9 341 (3) Keremedis/Tachtsis [2001] (N58T)