Point in Discrete Space is Adherent Point

Theorem

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

Let $U \subseteq S$.

Then $x$ is an adherent point of $U$ if and only if $x \in U$.

Proof

Let $U \subseteq S$.

From Set in Discrete Topology is Clopen it follows that $U$ is open in $T$.

Let $x \in U$.

Then $U$ is an open neighborhood of $x$ containing (trivially) an element of $U$, that is, $x$.

So, by definition, $x$ is an adherent point of $U$.

Now suppose $x \notin U$.

Then $\set x$ is an open neighborhood of $x$ such that $\set x \cap U = \O$.

So $x$ can not be an adherent point of $U$.

$\blacksquare$

Sources

• 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: $3$