Let us determine all topologies on a set X={a,b,c}:
τ1={∅,X}
τ2={∅,{a},{a,b},{a,c},X}
τ3={∅,{b},{a,b},{b,c},X}
τ4={∅,{c},{b,c},{a,c},X}
τ5={∅,{a},{b,c},X}
τ6={∅,{b},{a,c},X}
τ7={∅,{c},{a,b},X}
τ8={∅,{a},{b},{a,b},X}
τ9={∅,{b},{c},{b,c},X}
τ10={∅,{a},{c},{a,c},X}
τ11={∅,{a},{b},{c},{a,b},{a,c},{b,c},X}
τ12={∅,{a,b},X}
τ13={∅,{a,c},X}
τ14={∅,{b,c},X}
Note that if the topology contains two different 2-element open sets, then by definition it contains also their intersection, that is a singleton.
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