
theorem
  for T being anti-discrete non empty TopStruct, p being Point of T, D
  being Basis of p holds D = {the carrier of T}
proof
  let T be anti-discrete non empty TopStruct, p be Point of T, D be Basis of p;
  thus D c= {the carrier of T}
  proof
    let a be object;
    assume
A1: a in D;
    then reconsider X = a as Subset of T;
    X is open & p in X by A1,YELLOW_8:12;
    then X = the carrier of T by TDLAT_3:18;
    hence thesis by TARSKI:def 1;
  end;
  hence thesis by ZFMISC_1:33;
end;
