reserve n for Nat,
        X for set,
        Fx,Gx for Subset-Family of X;
reserve TM for metrizable TopSpace,
        TM1,TM2 for finite-ind second-countable metrizable TopSpace,
        A,B,L,H for Subset of TM,
        U,W for open Subset of TM,
        p for Point of TM,

        F,G for finite Subset-Family of TM,
        I for Integer;
reserve u for Point of Euclid 1,
  U for Point of TOP-REAL 1,
  r,u1 for Real,
  s for Real;

theorem Th19:
  for T be TopSpace,A be countable Subset of T st T|A is T_4
    holds A is finite-ind & ind A<=0
proof
  let T be TopSpace,A be countable Subset of T such that
A1: T|A is T_4;
  per cases;
  suppose A={}T;
    hence thesis by TOPDIM_1:6;
  end;
  suppose
A2: A is non empty;
    then reconsider TT=T as non empty TopSpace;
    reconsider a=A as non empty Subset of TT by A2;
    set Ta=TT|a;
    deffunc F(Point of Ta)={$1};
    defpred P[set] means
    not contradiction;
    defpred PP[set] means
    $1 in A & P[$1];
    consider S be Subset-Family of Ta such that
A3: S={F(w) where w is Point of Ta:PP[w]} from LMOD_7:sch 5;
    for B be Subset of Ta st B in S holds B is finite-ind & ind B<=0
    proof
      let B be Subset of Ta;
      assume B in S;
      then consider w be Point of Ta such that
A4:   B=F(w) and
      PP[w] by A3;
      card F(w)=1 by CARD_1:30;
      then ind F(w)<1+0 by TOPDIM_1:8;
      hence thesis by A4,INT_1:7;
    end;
    then
A5: S is finite-ind & ind S<=0 by TOPDIM_1:11;
    [#]Ta c=union S
    proof
      let x be object;
      assume
A6:   x in [#]Ta;
      then x in a by PRE_TOPC:def 5;
      then x in {x} & {x} in S by A3,A6,TARSKI:def 1;
      hence thesis by TARSKI:def 4;
    end;
    then
A7: S is Cover of Ta by SETFAM_1:def 11;
A8: S is closed
    proof
      let B be Subset of Ta;
      assume B in S;
      then ex w be Point of Ta st B=F(w) & PP[w] by A3;
      hence thesis by A1;
    end;
A9: card A c=omega by CARD_3:def 14;
    card{F(w) where w is Point of TT|a:PP[w]}c=card A from BORSUK_2:sch 1;
    then card S c=omega by A3,A9;
    then
A10: S is countable;
    [#]Ta=A by PRE_TOPC:def 5;
    then
A11: Ta is countable by ORDERS_4:def 2;
    then Ta is finite-ind by A1,A5,A7,A8,A10,TOPDIM_1:36;
    then
A12: A is finite-ind by TOPDIM_1:18;
    ind Ta<=0 by A1,A5,A7,A8,A10,A11,TOPDIM_1:36;
    hence thesis by A12,TOPDIM_1:17;
  end;
end;
