**2.13.4**

Problem:

- If is a family of topologies on , show that is a topology on . Is a topology on ?
- Let be a family of topologies on . Show that there is a unique smallest topology on containing all collections , and a unique largest topology contained in all .
- If , let and . Find the smallest topology containing and , and the largest topology contained in and .

Answer:

- is a topology on due to the fact that each is a topology on and if is empty, then is the discrete topology on . For , let and define the topologies and . If we take an intersection of and , then we are presented with a situation where . Therefore showing that is not a topology on .
- We must note that there exists a topology on containing all the collections (discrete topology). Thus, there exists a topology on contained in all . Therefore denoted as to be the largest topology on . For the smallest topology on that contains all the collections , we take the intersection of all topologies that satisfy the property of containing the union of all the such that where there exists at least one . Therefore I claim that is the largest topology and is the smallest topology on . To show the uniqueness of , let be a topology that contains all the . is the intersection of and therefore . Then suppose fits the criteria of , then and . And therefore , thus is unique. The uniqueness proof of is similar.
- The smallest topology containing and is the union of and such that for which has to be thrown in where . The largest topology contained in and is the intersection such that .

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