Ultraconnected Space is T4

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Theorem

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


Then $T$ is a $T_4$ space.


Proof

Recall the definition of a $T_4$ space:

$T = \left({S, \tau}\right)$ is a $T_4$ space if and only if:

for any two disjoint closed sets $A, B \subseteq S$, there exist disjoint open sets $U, V \in \tau$ containing $A$ and $B$ respectively.


As no two closed sets of an ultraconnected space are actually disjoint, it follows that $T_4$-ness follows vacuously.

$\blacksquare$


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