2. Let T be a subset family of nonempty set X. ..

3. If T meets the following conditions: both X and empty set belong to T. ..

4. the intersection of any two members in t belongs to t.

5. the union of any number of members in t belongs to t.

6. then t is called the topology on X.

7. The set x with topology t is called topological space, and is denoted as (x, t).

8. Let T 1 and T2 be two topologies on set X. ..

9. If T 1T2 is relevant, it means that T 1 is thicker than T2, or T2 is thinner than T 1.

10. When two topologies on X have nothing to do with each other, they are said to be incomparable.

1 1. On the set X, the discrete topology is the thinnest topology and the trivial topology is the coarsest topology.