
theorem Th31:
  for T being TopSpace, B being Basis of T for A being Subset of T holds
  A is open iff for p being Point of T st p in A
  ex a being Subset of T st a in B & p in a & a c= A
proof
  let T be TopSpace, K be Basis of T, A be Subset of T;
  hereby
    assume A is open;
    then
A1: A = union {G where G is Subset of T: G in K & G c= A} by YELLOW_8:9;
    let p be Point of T;
    assume p in A;
    then consider Z being set such that
A2: p in Z and
A3: Z in {G where G is Subset of T: G in K & G c= A} by A1,TARSKI:def 4;
    ex a being Subset of T st Z = a & a in K & a c= A by A3;
    hence ex a being Subset of T st a in K & p in a & a c= A by A2;
  end;
  assume
A4: for p being Point of T st p in A
  ex a being Subset of T st a in K & p in a & a c= A;
  set F = {G where G is Subset of T: G in K & G c= A};
  F c= bool the carrier of T
  proof
    let x be object;
    assume x in F;
    then ex G being Subset of T st x = G & G in K & G c= A;
    hence thesis;
  end;
  then reconsider F as Subset-Family of T;
  reconsider F as Subset-Family of T;
A5: F is open
  proof
    let x be Subset of T;
    assume x in F;
    then
A6: ex G being Subset of T st x = G & G in K & G c= A;
    K c= the topology of T by TOPS_2:64;
    hence x in the topology of T by A6;
  end;
  A = union F
  proof
    hereby
      let x be object;
      assume
A7:   x in A;
      then reconsider p = x as Point of T;
      consider a being Subset of T such that
A8:   a in K and
A9:   p in a and
A10:  a c= A by A4,A7;
      a in F by A8,A10;
      hence x in union F by A9,TARSKI:def 4;
    end;
    F c= bool A
    proof
      let x be object;
      assume x in F;
      then ex G being Subset of T st x = G & G in K & G c= A;
      hence thesis;
    end;
    then union F c= union bool A by ZFMISC_1:77;
    hence thesis by ZFMISC_1:81;
  end;
  hence thesis by A5,TOPS_2:19;
end;
