*Definition [#g5af0697]
-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 [#r50fdd83]
-A set S is said to be
++I-finite  if every nonvoid family of subsets of S has an &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5csubseteq%20%5c%5d%7d%25.png);-maximal element,
++I_a-finite if it is not the union of two disjoint sets neither of which is finite according to definition I,
++II-finite if every non-void &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5csubseteq%20%5c%5d%7d%25.png);-monotone familiy has a &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5csubseteq%20%5c%5d%7d%25.png);-maximal element,
++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)),
++IV-finite if it is irreflexive,
++V-finite if |S|=0 or 2. |S|>|S|,
++VI-finite if |S|=0 or |S|=1 or |S|^2>|S|,
++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;
&attachref(./finite_0.JPG,nolink);
*Remark [#d0e52262]
-ZFC公理系とは限らない。

*Reference [#w426e110]
-Spisiak, Ladislav and  Vojtas, Peter, ''Dependences between definitions of finiteness.'', Czech. Math. J. 38(113), No.3, 389-397 (1988).