F-compact

Definition

• 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

• A set S is said to be
  1. I-finite if every nonvoid family of subsets of S has an -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 -monotone familiy has a -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

• ZFC公理系とは限らない。

Reference

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