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 $x$.
Pages in category "Equivalence of Definitions of Limit Point"
The following 4 pages are in this category, out of 4 total.