reconsider F = {({} X),([#] X)} as Subset-Family of X by MEASURE1:2;
take F ; :: thesis: F is topology-like
for a being Subset-Family of X st a c= F holds
union a in F

proof

let a be Subset-Family of X; :: thesis: ( a c= F implies union a in F )
assume a c= F ; :: thesis: union a in F
then ( a = {} or a = {X} or a = {{},X} or a = {{}} ) by ZFMISC_1:36;
then ( union a = {} or union a = X or union a = {} \/ X or union a = {} ) by ZFMISC_1:75, ZFMISC_1:25, ZFMISC_1:2;
hence union a in F by TARSKI:def 2; :: thesis: verum

end;

then F is union-closed ;
hence F is topology-like by TARSKI:def 2; :: thesis: verum