let X be non empty TopSpace; :: thesis: for X1, X2 being non empty open SubSpace of X st X1 meets X2 holds
X1 meet X2 is open SubSpace of X
let X1, X2 be non empty open SubSpace of X; :: thesis: ( X1 meets X2 implies X1 meet X2 is open SubSpace of X )
assume A1: X1 meets X2 ; :: thesis: X1 meet X2 is open SubSpace of X
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
reconsider A = the carrier of (X1 meet X2) as Subset of X by Th1;
( A1 is open & A2 is open ) by Th16;
then A1 /\ A2 is open ;
then A is open by A1, Def4;
hence X1 meet X2 is open SubSpace of X by Th16; :: thesis: verum