# Open Extension Space is Ultraconnected

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## Theorem

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

Let $T^*_{\bar p} = \left({S^*_p, \tau^*_{\bar p}}\right)$ be the open extension space of $T$.

Then $T^*_{\bar p}$ is ultraconnected.

## Proof

Apart from $S^*_p$, every open set of $T^*_{\bar p}$ does not contain $p$, by definition of open extension space.

So, apart from $\varnothing$, every closed set of $T$ does contain $p$, by definition of closed set.

So every pair of closed sets of $T$ has an intersection which contains at least $p$.

So there are no non-empty disjoint closed sets of $T$.

Hence the result, by definition of ultraconnected.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 16: \ 9$