Definition:Convergent Sequence/Topology/Definition 1

Definition

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 $\alpha \in S$ if and only if:

$\forall U \in \tau: \alpha \in U \implies \paren {\exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in U}$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $4$: The Hausdorff condition: $4.1$: Motivation: Definition $4.1.2$