reserve Y for TopStruct;
reserve X for non empty TopSpace;

theorem Th38:
  for X being anti-discrete non empty TopSpace, A being non empty
  Subset of X holds A is discrete iff A is trivial
proof
  let X be anti-discrete non empty TopSpace, A be non empty Subset of X;
  hereby
    consider a being object such that
A1: a in A by XBOOLE_0:def 1;
    reconsider a as Point of X by A1;
    assume A is discrete;
    then consider G being Subset of X such that
A2: G is open and
A3: A /\ G = {a} by A1,Th26;
    G <> {} by A3;
    then
A4: G = the carrier of X by A2,TDLAT_3:18;
    now
      take a;
      thus A = {a} by A3,A4,XBOOLE_1:28;
    end;
    hence A is trivial;
  end;
  hereby
    assume A is trivial;
    then ex a being Element of A st A = {a} by SUBSET_1:46;
    hence A is discrete by Th30;
  end;
end;
