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 $A \setminus \set \alpha$.

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
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): limit point (accumulation point, cluster point): 1. (of a sequence)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): limit point (accumulation point, cluster point): 1. (of a sequence)