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

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Theorem

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

Let $\mathcal C$ be the open cover of $T$ defined as:

$\mathcal C = \left\{{\left\{{x, p}\right\}: x \in S, x \ne p}\right\}$


Then $\mathcal C$ has no open refinement which is locally finite.


Proof

Suppose $T$ is an infinite particular point space.

As $S$ is infinite, $\mathcal C$ is also infinite.


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

Then any neighborhood of $x$ must contain $p$, by the nature of the particular point topology.

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

That is:

$\forall C \in \mathcal C: p \in C$

So any neighborhood of $x$ intersects with all elements of the open cover $\mathcal C$.

As $\mathcal C$ is infinite, it therefore has no open refinement which is locally finite.

$\blacksquare$


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