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Add the following items to the bibliography:
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Kolany, A.
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[1999] Rad\`o selection lemma and other combinatorial statements uniformly
proved, preprint.
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Herrlich, H./Keremedis, K.
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[1998] Products, the Baire category theorem, and the axiom of dependent choice,
preprint October 12.
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Add Goldblatt [1985] as a reference for form [345 A].
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In form [43 D] add the word "filter": "... there is a ${\Cal D}$ generic
filter $G$ with $p\in G$."
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Forms [14 CM] and [43 W] through [43 AC] have been added. Also, the first
definition in note 28 has been revised. For these additions and revisions
choose one of the two below:
Put [14 CM] in CHOICE FORMS, V. Conditional Choice and put [43 W] through
[43 AC] in TOPOLOGICAL FORMS I. Baire Category Type Theorems and in II. Product
Theorems.
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In the bibliography, change Banaschewski [1998] to:
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[1998] Choice functions and compactness conditions , Math. Logic Quart.
44, 427-430.
change Felgner, U./Truss, J. K. [1999] to:
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[1999] The independence of the prime ideal theorem from the order extension
theorem , Fund. Math. 64, 199-215.
and change Keremedis, K./Tachtsis, E. [1999] to:
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Howard, P./Keremedis, K./Rubin, J./Stanley, A./Tachtsis E.,
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[1999] Non-constructive properties of the real numbers , preprint.
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Add the following items to the bibliography:
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Keremedis, K./Tachtsis, E.
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[1999a] On Loeb and weakly Loeb Hausdorf spaces preprint.
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Keremedis, K./Tachtsis, E.
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[1999b] The countable axiom of choice for finite sets does not imply compact
metric spaces are separable preprint.
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Add the two forms:
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[5 A] Partial Choice for Countable Families of Countable Sets of Reals: Every
countable family of non-empty countable sets of real numbers has an infinite
subset with a choice function. (See the proof of the equivalence of 94 and
[94 N].)
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FORM 338. $UT(\aleph_0,\aleph_0,WO)$: The union of a denumerable number of
denumerable sets is well orderable. Note 4.
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Add [5 A] to CHOICE FORMS, Part IV, Partial Choice and add form 338 to CARDINAL
NUMBER FORMS, Part III Cardinality of Unions.
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Form [8 F] should be a new form, form 339. This leaves [8 F] blank.
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FORM 339. Martin's Axiom $(\aleph_{0})$: Whenever $(P\le)$ is a non-empty,
ccc quasi-order (ccc means every anti-chain is countable) and ${\Cal D}$
is a family of $\le\aleph_0$ dense subsets of $P$, then there is a ${\Cal
D}$ generic filter $G$ in $P$. Kunen [1980], Shannon [1990], and note 47.
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Remove [8 F] and add form 339 to ORDERING RELATIONS, Part II, Versions of
Martin's Axiom.
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It was shown in Howard/Keremedis/Rubin/Stanley [1999] that forms 6 and 36
are equivalent. Thus, omit form 36 and add form [6 C]:
[6 C] If $A\subseteq{\Bbb R}^n$ and $A\bigcap B$ is countable for
every bounded $B$ then $A$ is countable. G. Moore [1982] p 36,
Keremedis/Howard/Rubin/Stanley/Tachtsis [1999].
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Add a new form 36:
FORM 36. Compact T$_2$ spaces are Loeb. (A space is Loeb if
the set of non-empty closed sets has a choice function.) Keremedis/Tachtsis
[1999a].