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;

theorem Th35:
  for T st T is T_1 & T is Lindelof holds T is finite-ind & ind T
  <= 0 iff for A,B be closed Subset of T st A misses B holds {}T separates A,B
proof
  let T such that
A1: T is T_1 and
A2: T is Lindelof;
  hereby
    assume
A3: T is finite-ind & ind T<=0;
    let A,B be closed Subset of T;
    assume A misses B;
    then consider A9,B9 be closed Subset of T such that
A4: A9 misses B9 and
A5: A9\/B9=[#]T and
A6: A c=A9 & B c=B9 by A2,A3,Th34;
A7: A9`=B9 by A4,A5,PRE_TOPC:5;
    (A9\/B9)`=({}T)`` & A9=B9` by A4,A5,PRE_TOPC:5;
    hence {}T separates A,B by A4,A6,A7,METRIZTS:def 3;
  end;
  assume
A8: for A,B be closed Subset of T st A misses B holds{}T separates A,B;
  for A,B be closed Subset of T st A misses B ex A9,B9 be closed Subset
  of T st A9 misses B9 & A9\/B9=[#]T & A c=A9 & B c=B9
  proof
    let A,B be closed Subset of T;
    assume A misses B;
    then {}T separates A,B by A8;
    then consider U,W be open Subset of T such that
A9: A c=U & B c=W and
A10: U misses W and
A11: {}T=(U\/W)` by METRIZTS:def 3;
A12: [#]T=(U\/W)`` by A11
      .=U\/W;
    then U=W` & W=U` by A10,PRE_TOPC:5;
    hence thesis by A9,A10,A12;
  end;
  hence thesis by A1,Th32;
end;
