Limit Point/Examples

Examples of Limit Points

End Points of Real Interval

The real number $a$ is a limit point of both the open real interval $\openint a b$ as well as of the closed real interval $\closedint a b$.

It is noted that $a \in \closedint a b$ but $a \notin \openint a b$.

Union of Singleton with Open Real Interval

Let $\R$ be the set of real numbers.

Let $H \subseteq \R$ be the subset of $\R$ defined as:

$H = \set 0 \cup \openint 1 2$

Then $0$ is not a limit point of $H$.

Real Number is Limit Point of Rational Numbers in Real Numbers

Let $\R$ be the set of real numbers.

Let $\Q$ be the set of rational numbers.

Let $x \in \R$.

Then $x$ is a limit point of $\Q$.

Zero is Limit Point of Integer Reciprocal Space

Let $A \subseteq \R$ be the set of all points on $\R$ defined as:

$A := \set {\dfrac 1 n : n \in \Z_{>0} }$

Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.

Then $0$ is the only limit point of $A$ in $\R$.