Definition:Limit of Sequence/Topological Space

Definition
Let $T = \left({S, \vartheta}\right)$ be a topological space.

Let $A \subseteq S$.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $A$.

Let $\left \langle {x_n} \right \rangle$ converge to a value $\alpha \in A$.

Then $\alpha$ is known as a limit (point) of $\left \langle {x_n} \right \rangle$ (as $n$ tends to infinity).

Some sources insist that $\left \langle {x_n} \right \rangle$ be a sequence in $A \setminus \left\{{\alpha}\right\}$.