Cover of Doubletons of Infinite Particular Point Space has no Locally Finite Refinement

Theorem
Let $T = \left({S, \vartheta_p}\right)$ be an infinite particular point space.

Let $\mathcal C$ be the open cover of $T$ defined as:
 * $\mathcal C = \left\{{\left\{{x, p}\right\}: x \in S, x \ne p}\right\}$

has no refinement which is locally finite.

Proof
Suppose $T$ is an infinite particular point space.

The cardinality of the open cover $\mathcal C$ is $\left|{S \setminus \left\{{p}\right\}}\right|$, that is,

Trivially, any refinement of $\mathcal C$ which is an open cover must be $\mathcal C$ itself.

Let $x \in X, x \ne p$.

Then any neighborhood of $x$ must contain $p$.

But $p$ is contained in all elements of $\mathcal C$.

So any neighborhood of $x$ must intersect with an infinite number of elements of $\mathcal C$.

So, by definition, $\mathcal C$ has no refinement which is locally finite.