Definition:Limit Point/Topology

Definition
Let $T = \left({S, \vartheta}\right)$ be a topological space.

Limit Point of Point
The concept of a limit point can be sharpened to apply to individual points, as follows:

Definition from Neighborhood
Some sources define a point $x \in S$ to be a limit point of $A$ if every neighborhood $U$ of $x$ satisfies $A \cap \left({U \setminus \left\{{x}\right\}}\right) \ne \varnothing$.

This definition is the same as the definition of a limit point of a set if the definition of neighborhood is the one which insists that the neighborhood must be open in $T$.

Simple Examples

 * $0$ is the only limit point of the set $\left\{{1/n: n \in \N}\right\}$ in the usual topology of $\R$.


 * Every point of $\R$ is a limit point of $\R$ in the usual topology.


 * In $\R$ under the usual topology, $a$ is a limit point of the open interval $\left({a . . b}\right)$ and also of the closed interval $\left[{a . . b}\right]$. Thus it can be seen that a limit point of a set may or may not be part of that set.


 * From Rationals Dense in Reals, it is shown that any point $x \in \R$ is a limit point of the set of rational numbers $\Q$. This is an interesting case, because $\Q$ is countable but its set of limit points in $\R$ is $\R$ itself, which is uncountable.


 * The set $\Z$ has no limit points in the usual topology of $\R$.