let X0 be SubSpace of X; :: thesis: ( X0 is boundary & X0 is closed implies X0 is nowhere_dense )
assume A1: ( X0 is boundary & X0 is closed ) ; :: thesis: X0 is nowhere_dense

now :: thesis: for A being Subset of X st A = the carrier of X0 holds
A is nowhere_dense

let A be Subset of X; :: thesis: ( A = the carrier of X0 implies A is nowhere_dense )
assume A = the carrier of X0 ; :: thesis: A is nowhere_dense
then ( A is boundary & A is closed ) by A1, TSEP_1:11;
hence A is nowhere_dense ; :: thesis: verum

end;

hence X0 is nowhere_dense ; :: thesis: verum