\begin{definition}[Definition:Convergent Sequence/Topology/Definition 1]
Let $T = \struct {S, \tau}$ be a topological space.
Let $A \subseteq S$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $A$.
Then $\sequence {x_n}$ '''converges to the limit $\alpha \in S$''' {{iff}}:
:$\forall U \in \tau: \alpha \in U \implies \paren {\exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in U}$
\end{definition}