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118 A] For each complete linear order, the set of non-empty convex subsets has a choice function, van Douwen [1985]. [118 B] For each complete linear order, each pairwise disjoint collection of non-empty, convex, open subsets has a choice function, van Douwen [1985]. F O R M 118: LINEARLY O R D E R E D SPACES A R E N O R M A L 43 [118 C] Every linearly orderable space is collectionwise normal, van Douwen [1985] and note 71. [118 D] Every linearly orderable space is collectionwise Hausdorff. van Douwen [1985] and note 71.

MCUJ(OC,OO) (CJ-MC): For every family X of pairwise disjoint nonempty sets, there is a function / such that for each x G X, f(x) is a non-empty countable subset of x. Howard/Keremedis/Rubin/Rubin [1997b]. [76 A] Every open cover U of a metric space can be written as a well ordered union (J{E/a, • ex G 7} where 7 is an ordinal and each Ua is locally countable. Howard/Keremedis/Rubin/Stanley [1997] and note 141. [76 B] Every open cover U of a metric space can be written as a well ordered union (J{Z7a : a G 7} where 7 is an ordinal and each Ua is point countable.

V°{X) = X and Vn+l(X) = V{Vn{X)). (K(0) is equivalent to form 1 (AC) and K(l) is equivalent to the selection principle (form 15)). Monro [1972], [1973b] and note 90. FORM 82. E(IJII) (Howard/Yorke [1989]): If X is infinite then V{X) is Dedekind infinite. ) Levy [1958] and note 94. [82 A]. Every infinite set can be mapped onto UJ. Howard/Yorke [1989], Monro [1972]. [82 B] E(Ia,III): For every set X, if T{X) is Dedekind finite then X is amorphous. Howard/Yorke [1989], notes 57 and 94. FORM 83. E(IJI) (Howard/Yorke [1989]): T-finite is equivalent to finite.

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A Theory of Ordinary p-Adic Curves by Mochizuki S.


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