set F = {({} T)};
{({} T)} c= bool the carrier of T

proof

let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in {({} T)} or a in bool the carrier of T )
assume a in {({} T)} ; :: thesis: a in bool the carrier of T
then a = {} T by TARSKI:def 1;
hence a in bool the carrier of T ; :: thesis: verum

end;

then reconsider F = {({} T)} as Subset-Family of T ;
take F ; :: thesis: ( not F is empty & F is mutually-disjoint & F is open & F is closed )
thus ( not F is empty & F is mutually-disjoint ) by TAXONOM2:10; :: thesis: ( F is open & F is closed )
thus F is open by TARSKI:def 1; :: thesis: F is closed
let P be Subset of T; :: according to TOPS_2:def 2 :: thesis: ( not P in F or P is closed )
thus ( not P in F or P is closed ) by TARSKI:def 1; :: thesis: verum