Particular Point Space is not Ultraconnected
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Theorem
Let $T = \struct {S, \tau_p}$ 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 $\set x$ and $\set y$.
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 $\set x \cap \set y = \O$.
The result follows by definition of ultraconnected.
$\blacksquare$
Also see
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $8 \text { - } 10$. Particular Point Topology: $10$