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 of $\sequence {x_n}$.

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

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.

$\blacksquare$

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1 \text { - } 3$. Discrete Topology: $3$