
theorem
  for T being non empty TopSpace, A being Subset of T, p being Point of
  T holds p in Fr A iff for B being Basis of p, U being Subset of T st U in B
  holds A meets U & U \ A <> {}
proof
  let T be non empty TopSpace, A be Subset of T, p be Point of T;
  hereby
    assume
A1: p in Fr A;
    let B be Basis of p, U be Subset of T;
    assume U in B;
    then U is open & p in U by YELLOW_8:12;
    hence A meets U & U \ A <> {} by A1,Th9;
  end;
  assume
A2: for B being Basis of p, U being Subset of T st U in B holds A meets
  U & U \ A <> {};
  for U being Subset of T st U is open & p in U holds A meets U & U meets A`
  proof
    set B = the Basis of p;
    let U be Subset of T;
    assume U is open & p in U;
    then consider V being Subset of T such that
A3: V in B and
A4: V c= U by YELLOW_8:def 1;
    V \ A <> {} by A2,A3;
    then
A5: U \ A <> {} by A4,XBOOLE_1:3,33;
    A meets V by A2,A3;
    hence thesis by A4,A5,Th1,XBOOLE_1:63;
  end;
  hence thesis by TOPS_1:28;
end;
