Definition:Convergent Sequence/Topology

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 $A$.

Definition 1

Then $\sequence {x_n}$ converges to the limit $\alpha \in S$ if and only if:

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

Definition 2

Then $\sequence {x_n}$ converges to the limit $\alpha \in S$ if and only if:

$\forall U \in \tau: \alpha \in U \implies \set {n \in \N: x_n \notin U}$ is finite.

Such a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $A$ is said to be convergent in $S$ or simply convergent when $S$ is understood.

Note on Domain of $N$

Some sources insist that $N \in \N$ but this is not strictly necessary and can make proofs more cumbersome.

Also see

• Equivalence of Definitions of Convergent Sequence in Topology

• Results about convergent sequences in the context of topology can be found here.