reserve a,b,c for set;

theorem Th15:
  for T being finite-weight non empty TopSpace ex B0 being Basis
of T st ex f being Function of the carrier of T, the topology of T st B0 = rng
  f & for x being Point of T holds x in f.x & for U being open Subset of T st x
  in U holds f.x c= U
proof
  let T be finite-weight non empty TopSpace;
  consider B being Basis of T such that
A1: card B = weight T by WAYBEL23:74;
  deffunc F(object) = meet {U where U is Subset of T: $1 in U & U in B};
A2: B is finite by A1,Def4;
A3: now
    B c= the topology of T by TOPS_2:64;
    then FinMeetCl B c= FinMeetCl the topology of T by CANTOR_1:14;
    then
A4: FinMeetCl B c= the topology of T by CANTOR_1:5;
    let a be object;
    assume a in the carrier of T;
    then reconsider x = a as Point of T;
    set S = {U where U is Subset of T: x in U & U in B};
    consider U being Subset of T such that
A5: U in B and
A6: x in U and
    U c= [#]T by YELLOW_9:31;
A7: S c= B
    proof
      let a be object;
      assume a in S;
      then ex U being Subset of T st a = U & x in U & U in B;
      hence thesis;
    end;
    then reconsider S as Subset-Family of T by XBOOLE_1:1;
    Intersect S in FinMeetCl B by A2,A7,CANTOR_1:def 3;
    then
A8: Intersect S in the topology of T by A4;
    U in S by A5,A6;
    hence F(a) in the topology of T by A8,SETFAM_1:def 9;
  end;
  consider f being Function of the carrier of T, the topology of T such that
A9: for a being object st a in the carrier of T holds f.a = F(a)
from FUNCT_2:sch 2(
  A3);
  set B0 = rng f;
  reconsider B0 as Subset-Family of T by XBOOLE_1:1;
A10: now
    let a;
    assume a in the carrier of T;
    then reconsider x = a as Point of T;
    set S = {U where U is Subset of T: x in U & U in B};
    consider U being Subset of T such that
A11: U in B and
A12: x in U and
    U c= [#]T by YELLOW_9:31;
A13: now
      let b;
      assume b in S;
      then ex U being Subset of T st b = U & a in U & U in B;
      hence a in b;
    end;
A14: f.a = meet S by A9;
    U in S by A11,A12;
    hence a in f.a by A14,A13,SETFAM_1:def 1;
  end;
A15: now
    let x be Point of T, V be Subset of T;
    set S = {U where U is Subset of T: x in U & U in B};
    assume that
A16: x in V and
A17: V is open;
    consider U being Subset of T such that
A18: U in B and
A19: x in U and
A20: U c= V by A16,A17,YELLOW_9:31;
A21: f.x = meet S by A9;
    U in S by A18,A19;
    then f.x c= U by A21,SETFAM_1:3;
    hence f.x c= V by A20;
  end;
  now
    let A be Subset of T;
    assume
A22: A is open;
    let p be Point of T;
    assume
A23: p in A;
    f.p in the topology of T;
    then reconsider a = f.p as Subset of T;
    take a;
    thus a in B0 by FUNCT_2:4;
    thus p in a by A10;
    thus a c= A by A15,A22,A23;
  end;
  then reconsider B0 as Basis of T by YELLOW_9:32;
  take B0,f;
  thus B0 = rng f;
  let x be Point of T;
  thus thesis by A10,A15;
end;
