Singleton Point is Isolated

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Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $x \in S$.


Then $x$ is an isolated point of the singleton set $\left\{{x}\right\}$, but not necessarily an isolated point of $T$.


Proof

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

The fact that such a $U$ exists follows from the fact that $S$ is open in $T$ and $x \in S$.

Then as $\left\{{x}\right\} \subseteq S$ it follows that $\left\{{x}\right\} \cap U = \left\{{x}\right\}$ from Intersection with Subset is Subset‎.

So by definition, $x$ is an isolated point of $\left\{{x}\right\}$.


From Topological Space is Discrete iff All Points are Isolated, unless $T$ is the discrete space on $S$, not all elements of $T$ are isolated points of $T$.

$\blacksquare$