Non-Trivial Particular Point Topology is not T3

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Theorem

Let $T = \left({S, \tau_p}\right)$ be a particular point space such that $S$ is not a singleton.


Then $T$ is not a $T_3$ space.


Proof

We have that there are at least two distinct elements of $S$.

So, consider $x, p \in S: x \ne p$.

Then $X = \left\{{x}\right\}$ is closed in $T$ and $p \notin X$.

Suppose $U \in \tau_p$ is an open set in $T$ such that $X \subseteq U$.

We have that $\left\{{p}\right\} \in \tau_p$ such that $p \in \left\{{p}\right\}$.

But as $p \in U, p \in \left\{{p}\right\}$ we have that $U \cap \left\{{p}\right\} \ne \varnothing$.

So $T$ is not a $T_3$ space.

$\blacksquare$


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