Ok, I was just wondering. They say in my book to think of a strictly finer topology of T to be the same as T but some of the sets in T are divided into more peices. But can't you have a strictly finer topology of T that consists of the same elements as T but the finer topology has 1 set that properly contains a set in T (Its only intersection would be the sets it properly contains and of course X, the topological space). Also, more than 1 set with this property should also work to make it a stricly finer topology than T. So if this works, then can't you think of a stricly finer topology of T as being one that contains T but also has "bigger sets" instead of the "finer sets"? Note: I added an example in the picture attachment. All sets in black are T (plus their unions and intersections). The blue set is a set of the finer topology.