Definition:Topology

Definition 2
Some sources include the additional assumption that $\O \in \tau$.

This is redundant as established in Empty Set is Element of Topology, since $(\text O 2)$ and $(\text O 1')$ are read to include the union over the empty set, which is equal to $\O$.

Also see

 * Equivalence of Definitions of Topology


 * $\sigma$-algebra, which looks similar on the surface to a topology, but closed (in the algebraic sense) under countable unions. A topology has no such limitation on countability.