Category:Equivalence of Definitions of Limit Point

This category contains pages concerning Equivalence of Definitions of Limit Point:

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

Let $A \subseteq S$.

The following definitions of the concept of limit point are equivalent:

Definition 1

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 2

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 3

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 4

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 $x$.