Given a set X={a,b,c} is the following subset a topology for X:
{ X,{a},{b},{a,b},{a,c},{empty set} } ?
Does it satisfy the criteria for a topology?
1) Does it include the empty set?
Yes.
2) Does it include X itself?
Yes.
3) If it includes set U and V does it include $\displaystyle U\cap V$.
Since there are only 6 sets here, you could just check every possible intersection- yes, it does,.
4) Are all possible unions of members of the collection in the collection?
Again, there are only a finite number of unions to try- yes, it does.
So this is, in fact, a topology for X.