Courtesy Wiki:

I dont know topology but I think proving 2 and 3 is easy. Since they follow directly from set properties...A subset Σ of the power set of a setXis a σ-algebra if and only if it has the following properties:

- Σ is nonempty
- If
Eis in Σ then so is the complement (X\E) ofE.- The union of countably many sets in Σ is also in Σ.

Which axiom are you having trouble with?