Limit Point of Sequence in Discrete Space not always Limit Point of Open Set

Theorem
Let $S$ be a set.

Let $T = \left({S, \tau}\right)$ be the discrete space on $S$.

Let $U \in \tau$ be an open set of $T$.

Let $\left\langle{x_n}\right\rangle$ be a sequence in $U$.

Let $x$ be the limit point of $\left\langle{x_n}\right\rangle$.

Then $x$ is not always a limit point of $U$.

Proof
Let $T = \left({S, \tau}\right)$ be the discrete space on $S$.

Let $x \in S$.

Then by definition of discrete space:
 * $U = \left\{{x}\right\}$ is an open set of $T$.

Consider the sequence $\left\langle{x_n}\right\rangle$ defined as:
 * $\forall n \in \N: x_n = x$

That is:
 * $\left\langle{x_n}\right\rangle = \left({x, x, x, \ldots}\right)$

Thus $x$ is the limit point of $\left\langle{x_n}\right\rangle$.

But:
 * $U \setminus \left\{{x}\right\} = \varnothing$

and so $x$ is not a limit point of $U$.

Hence the result.

Also see

 * Point in Discrete Space is Adherent Point