let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X st X1 is nowhere_dense & X2 is nowhere_dense holds
X1 union X2 is nowhere_dense
let X1, X2 be non empty SubSpace of X; :: thesis: ( X1 is nowhere_dense & X2 is nowhere_dense implies X1 union X2 is nowhere_dense )
reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;
assume ( X1 is nowhere_dense & X2 is nowhere_dense ) ; :: thesis: X1 union X2 is nowhere_dense
then ( A1 is nowhere_dense & A2 is nowhere_dense ) ;
then A1 \/ A2 is nowhere_dense by TOPS_1:53;
then for A being Subset of X st A = the carrier of (X1 union X2) holds
A is nowhere_dense by TSEP_1:def 2;
hence X1 union X2 is nowhere_dense ; :: thesis: verum