
theorem Th14:
  for T being non empty TopStruct holds T is compact iff for F
  being Subset-Family of T st F is open & [#](T) c= union(F) ex G being
  Subset-Family of T st G c= F & [#]T c= union G & G is finite
proof
  let T be non empty TopStruct;
  thus T is compact implies for F being Subset-Family of T st F is open & [#](
T) c= union(F) ex G being Subset-Family of T st G c= F & [#]T c= union G & G is
  finite
  proof
    assume
A1: T is compact;
    let F be Subset-Family of T;
    assume that
A2: F is open and
A3: [#](T) c= union(F);
    F is Cover of T by A3,TOPMETR:1;
    then consider G being Subset-Family of T such that
A4: G c= F & G is Cover of T & G is finite by A1,A2;
    take G;
    thus thesis by A4,TOPMETR:1;
  end;
  assume
A5: for F being Subset-Family of T st F is open & [#](T) c= union(F) ex
  G being Subset-Family of T st G c= F & [#]T c= union G & G is finite;
  let F be Subset-Family of T;
  assume that
A6: F is Cover of T and
A7: F is open;
  [#](T) c= union(F) by A6,TOPMETR:1;
  then consider G being Subset-Family of T such that
A8: G c= F & [#]T c= union G & G is finite by A5,A7;
  take G;
  thus thesis by A8,TOPMETR:1;
end;
