Definition of convergent sequence in topological space

The book I'm reading says:

If $\{x_n\}$ is a sequence of points in $(X,T)$ and $x \in (X,T)$, the sequence is said to converge to $x$ iff for each open set $U \in T$, there exists an $n_0 \in \Bbb N$ such that $x_n \in U$, for every $n \ge n_0$.

Is this definition wrong? Wouldn't $x$ have to be in $U$ for this to make sense? Otherwise you could just choose an open set that has no points of $x_n$ in it and it won't be convergent.

Shouldn't it be:

If $\{x_n\}$ is a sequence of points in $(X,T)$ and $x \in (X,T)$, the sequence is said to converge to $x$ iff for each open set $U \in T$ such that $x \in U$, there exists an $n_0 \in \Bbb N$ such that $x_n \in U$, for every $n \ge n_0$.

You're right. Another way to say this would be, "for each open neighbourhood $U$ of $x$, ...".