Definition:Limit Point/Topology/Set
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Definition
Let $T = \left({S, \tau}\right)$ be a topological space.
Let $A \subseteq S$.
Definition from Open Neighborhood
A point $x \in S$ is a limit point of $A$ if and only if every open neighborhood $U$ of $x$ satisfies:
- $A \cap \paren {U \setminus \set x} \ne \O$
That is, if and only if every open set $U \in \tau$ such that $x \in U$ contains some point of $A$ distinct from $x$.
Definition from Closure
A point $x \in S$ is a limit point of $A$ if and only if
- $x$ belongs to the closure of $A$ but is not an isolated point of $A$.
Definition from Adherent Point
A point $x \in S$ is a limit point of $A$ if and only if $x$ is an adherent point of $A$ but is not an isolated point of $A$.
Definition from Relative Complement
A point $x \in S$ is a limit point of $A$ if and only if $\left({S \setminus A}\right) \cup \left\{{x}\right\}$ is not a neighborhood of $x$.
Also see
- Results about limit points can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): limit point (accumulation point, cluster point): 2. (of a set)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): limit point (accumulation point, cluster point): 2. (of a set)