Definition:Accumulation Point/Sequence
< Definition:Accumulation Point(Redirected from Definition:Accumulation Point of Sequence)
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Definition
Let $\struct {S, \tau}$ be a topological space.
Let $A \subseteq S$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $A$.
Let $x \in S$.
Suppose that:
- $\forall U \in \tau: x \in U \implies \set {n \in \N: x_n \in U}$ is infinite
Then $x$ is an accumulation point of $\sequence {x_n}$.
Also see
- Results about accumulation points 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