reserve T,T1,T2 for TopSpace,
  A,B for Subset of T,
  F for Subset of T|A,
  G,G1, G2 for Subset-Family of T,
  U,W for open Subset of T|A,
  p for Point of T|A,
  n for Nat,
  I for Integer;
reserve Af for finite-ind Subset of T,
  Tf for finite-ind TopSpace;
reserve TM for metrizable TopSpace;

theorem Th39:
  for Null be Subset of TM st TM|Null is second-countable holds
Null is finite-ind & ind Null<=0 iff ex B be Basis of TM st for A be Subset of
  TM st A in B holds Null misses Fr A
proof
  set cTM=[#]TM;
  set TOP=the topology of TM;
  let Null be Subset of TM such that
A1: TM|Null is second-countable;
  hereby
    defpred P[object,object] means
for p be Point of TM,A be Subset of TM st$1=[p,A]
holds$2 in TOP & (not p in A implies $2={}TM) & (p in A implies ex W be open
    Subset of TM st W=$2 & p in W & W c=A & Null misses Fr W);
    assume
A2: Null is finite-ind & ind Null<=0;
A3: for x be object st x in [:cTM,TOP:]ex y be object st P[x,y]
    proof
      let x be object;
      assume x in [:cTM,TOP:];
      then consider p9,A9 be object such that
A4:   p9 in cTM and
A5:   A9 in TOP and
A6:   x=[p9,A9] by ZFMISC_1:def 2;
      reconsider p9 as Point of TM by A4;
      reconsider A9 as open Subset of TM by A5,PRE_TOPC:def 2;
      per cases;
      suppose
A7:     not p9 in A9;
        take{}TM;
        let p be Point of TM,A be Subset of TM such that
A8:     x=[p,A];
        p=p9 by A6,A8,XTUPLE_0:1;
        hence thesis by A6,A7,A8,PRE_TOPC:def 2,XTUPLE_0:1;
      end;
      suppose
        p9 in A9;
        then consider W be open Subset of TM such that
A9:     p9 in W & W c=A9 & Null misses Fr W by A1,A2,Th38;
        take W;
        let p be Point of TM,A be Subset of TM;
        assume x=[p,A];
        then p=p9 & A=A9 by A6,XTUPLE_0:1;
        hence thesis by A9,PRE_TOPC:def 2;
      end;
    end;
    consider f be Function such that
A10: dom f=[:cTM,TOP:] and
A11: for x be object st x in [:cTM,TOP:] holds P[x,f.x]
from CLASSES1:sch
    1( A3);
A12: rng f c=TOP
    proof
      let y be object;
      assume y in rng f;
      then consider x be object such that
A13:  x in dom f and
A14:  f.x=y by FUNCT_1:def 3;
      ex p,A be object st p in cTM & A in TOP & x=[p,A]
by A10,A13,ZFMISC_1:def 2;
      hence thesis by A10,A11,A13,A14;
    end;
    then reconsider RNG=rng f as Subset-Family of TM by XBOOLE_1:1;
    now
      let A be Subset of TM;
      assume A is open;
      then
A15:  A in TOP by PRE_TOPC:def 2;
      let p be Point of TM such that
A16:  p in A;
A17:  [p,A] in [:cTM,TOP:] by A15,A16,ZFMISC_1:87;
      then consider W be open Subset of TM such that
A18:  W=f.[p,A] & p in W & W c=A and
      Null misses Fr W by A11,A16;
      reconsider W as Subset of TM;
      take W;
      thus W in RNG & p in W & W c=A by A10,A17,A18,FUNCT_1:def 3;
    end;
    then reconsider RNG as Basis of TM by A12,YELLOW_9:32;
    take RNG;
    let B be Subset of TM;
    assume B in RNG;
    then consider x be object such that
A19: x in dom f and
A20: f.x=B by FUNCT_1:def 3;
    consider p,A be object such that
A21: p in cTM and
A22: A in TOP and
A23: x=[p,A] by A10,A19,ZFMISC_1:def 2;
    reconsider A as set by TARSKI:1;
    per cases;
    suppose
      p in A;
      then ex W be open Subset of TM st W=f.[p,A] & p in W & W c=A & Null
      misses Fr W by A10,A11,A19,A22,A23;
      hence Null misses Fr B by A20,A23;
    end;
    suppose
      not p in A;
      then B={}TM by A10,A11,A19,A20,A21,A22,A23;
      then Fr B={}TM by TOPGEN_1:14;
      hence Null misses Fr B;
    end;
  end;
  given B be Basis of TM such that
A24: for A be Subset of TM st A in B holds Null misses Fr A;
  for p be Point of TM,U be open Subset of TM st p in U ex W be open
  Subset of TM st p in W & W c=U & Null misses Fr W
  proof
    let p be Point of TM,U be open Subset of TM;
    assume p in U;
    then consider a be Subset of TM such that
A25: a in B and
A26: p in a & a c=U by YELLOW_9:31;
    B c=TOP by TOPS_2:64;
    then reconsider a as open Subset of TM by A25,PRE_TOPC:def 2;
    take a;
    thus thesis by A24,A25,A26;
  end;
  hence thesis by A1,Th38;
end;
