I have a worksheet for my real analysis course that sets up a problem by defining "a new topology $S$ on $\mathbb R$ by declaring a set $U \subseteq \mathbb R$ to be in $S$ if $\mathbb R \setminus U$ is finite." We haven't covered topologies in class yet, but the worksheet says that one of the conditions for a thing $T$ to be a topology is that $\emptyset \in T$.
It seems to me that $\mathbb R \setminus \emptyset$ would not be finite, because it would simply be $\mathbb R$ which is not finite. So $\emptyset \subset S$, but $\emptyset \notin S$. This would mean $S$ is not a topology. Have I misunderstood something about topologies?