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Last-modified: 2011-10-25 (火) 19:17:40

Definition Edit

  • A topological space X is called F-compact if every open cover of X contains of an F-finite subcover (F-finiteness stands for any reasonable definition of finiteness).

Finiteness Edit

  • A set S is said to be
    1. I-finite if every nonvoid family of subsets of S has an imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5csubseteq%20%5c%5d%7d%25.png-maximal element,
    2. I_a-finite if it is not the union of two disjoint sets neither of which is finite according to definition I,
    3. II-finite if every non-void imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5csubseteq%20%5c%5d%7d%25.png-monotone familiy has a imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5csubseteq%20%5c%5d%7d%25.png-maximal element,
    4. III-fnite if the power set of S is irreflexive (i.e., there is no one-to-one mapping of P(S) onto a proper subset of P(S)),
    5. IV-finite if it is irreflexive,
    6. V-finite if |S|=0 or 2. |S|>|S|,
    7. VI-finite if |S|=0 or |S|=1 or |S|^2>|S|,
    8. VII-finite if S is Ⅰ-finite or S is not well-orderable.
  • Let F very over I,I_a,II,III,IV,V,VI,VII. We say that a set S is F"-finite if P(S) is F-finite.
  • In ZF (or ZFU) the following implications are provable;

Remark Edit

  • ZFC公理系とは限らない。

Reference Edit

  • Spisiak, Ladislav and Vojtas, Peter, Dependences between definitions of finiteness., Czech. Math. J. 38(113), No.3, 389-397 (1988).