Point in Discrete Space is Adherent Point

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Theorem

Let $T = \left({S, \tau}\right)$ 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 $\left\{{x}\right\}$ is an open neighborhood of $x$ such that $\left\{{x}\right\} \cap U = \varnothing$.

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

$\blacksquare$


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