Definition:Limit of Sequence/Topological Space

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$.

Let $\sequence {x_n}$ be a sequence in $A$.

Let $\sequence {x_n}$ converge to a value $\alpha \in S$.

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

Also defined as

Some sources insist that $\sequence {x_n}$ be a sequence in $A \setminus \set \alpha$.

Also known as

A limit of $\sequence {x_n}$ as $n$ tends to infinity can also be presented more tersely as a limit of $\sequence {x_n}$ or even just limit of $x_n$.

Some sources present $\ds \lim_{n \mathop \to \infty} x_n$ as $\lim_n x_n$.

Also see

• Results about limits of sequences can be found here.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Limit Points