reserve x,y,X for set;

theorem
  for T being non empty TopSpace holds T is compact iff for N being net
  of T st N is Cauchy holds N is convergent
proof
  let T be non empty TopSpace;
  thus T is compact implies for N being net of T st N is Cauchy holds N is
  convergent
  proof
    assume
A1: T is compact;
    let N be net of T such that
A2: for A being Subset of T holds N is_eventually_in A or N
    is_eventually_in A`;
    consider x being Point of T such that
A3: x is_a_cluster_point_of N by A1,Lm1;
    now
      let V be a_neighborhood of x;
      assume not N is_eventually_in V;
      then N is_eventually_in V` by A2;
      then consider i being Element of N such that
A4:   for j being Element of N st i <= j holds N.j in V`;
      N is_often_in V by A3;
      then consider j being Element of N such that
A5:   i <= j and
A6:   N.j in V;
      N.j in V` by A4,A5;
      hence contradiction by A6,XBOOLE_0:def 5;
    end;
    then x in Lim N by YELLOW_6:def 15;
    hence thesis by YELLOW_6:def 16;
  end;
  assume
A7: for N being net of T st N is Cauchy holds N is convergent;
  now
    let F be ultra Filter of BoolePoset [#]T;
    set x = the Element of Lim a_net F;
    a_net F is convergent by A7;
    then
A8: Lim a_net F <> {} by YELLOW_6:def 16;
    then x in Lim a_net F;
    then reconsider x as Point of T;
    take x;
    thus x is_a_convergence_point_of F, T by A8,Th17;
  end;
  hence thesis by Th31;
end;
