reserve x,y,X for set;

theorem
  for T being non empty TopSpace holds T is compact iff for N being net
  of T ex S being subnet of N st S is convergent
proof
  let T be non empty TopSpace;
  hereby
    assume
A1: T is compact;
    let N be net of T;
    consider x being Point of T such that
A2: x is_a_cluster_point_of N by A1,Th33;
    consider S being subnet of N such that
A3: x in Lim S by A2,WAYBEL_9:32;
    take S;
    thus S is convergent by A3,YELLOW_6:def 16;
  end;
  assume
A4: for N being net of T ex S being subnet of N st S is convergent;
  now
    let N be net of T;
    consider S being subnet of N such that
A5: S is convergent by A4;
    set x = the Element of Lim S;
A6: Lim S <> {} by A5,YELLOW_6:def 16;
    then x in Lim S;
    then reconsider x as Point of T;
    take x;
    thus x is_a_cluster_point_of N by A6,WAYBEL_9:29,31;
  end;
  hence thesis by Th33;
end;
