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

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

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

Let $\sequence {x_n}$ be a sequence in $U$.

Let $x$ be the limit point of $\sequence {x_n}$.

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

Proof
Let $x \in S$.

By definition of discrete space:
 * $U = \set x$ is an open set of $T$.

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

That is:
 * $\sequence {x_n} = \tuple {x, x, x, \ldots}$

Thus $x$ is the limit point of $\sequence {x_n}$.

But:
 * $U \setminus \set x = \O$

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

Hence the result.

Also see

 * Point in Discrete Space is Adherent Point