
theorem Th15:
  for T being non empty TopSpace, B being prebasis of T holds T is
  compact iff for F being Subset of B st [#](T) c= union(F) ex G being finite
  Subset of F st [#]T c= union G
proof
  let T be non empty TopSpace, B be prebasis of T;
  set x = the carrier of T;
  the carrier of T in the topology of T by PRE_TOPC:def 1;
  then reconsider x as Element of InclPoset the topology of T by YELLOW_1:1;
  x is compact iff x << x by WAYBEL_3:def 2;
  hence thesis by WAYBEL_3:37,WAYBEL_7:31;
end;
