reserve X for non empty TopSpace,
  A,B for Subset of X;
reserve Y1,Y2 for non empty SubSpace of X;
reserve X1, X2 for non empty SubSpace of X;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;

theorem
  for X being non empty TopSpace holds (for X0 being SubSpace of X holds
  X0 is non boundary) implies X is discrete
proof
  let X be non empty TopSpace;
  assume
A1: for X0 being SubSpace of X holds X0 is non boundary;
  now
    let A be non empty Subset of X;
    consider X0 being strict non empty SubSpace of X such that
A2: A = the carrier of X0 by TSEP_1:10;
    X0 is non boundary by A1;
    hence A is not boundary by A2;
  end;
  hence thesis by TEX_1:def 5;
end;
