Particular Point Space is not Ultraconnected

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Theorem

Let $T = \left({S, \tau_p}\right)$ be a particular point space with at least three points.


Then $T$ is not ultraconnected.


Proof

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

Consider $\left\{{x}\right\}$ and $\left\{{y}\right\}$.

Neither are open as neither contain $p$.

So from Subset of Particular Point Space is either Open or Closed they are both closed.

We have that $\left\{{x}\right\} \cap \left\{{y}\right\} = \varnothing$.

The result follows by definition of ultraconnected.

$\blacksquare$


Also see


Sources