Ultraconnected Space is T4
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Theorem
Let $T = \struct {S, \tau}$ be a topological space which is ultraconnected.
Then $T$ is a $T_4$ space.
Proof
Recall the definition of a $T_4$ space:
$T = \struct {S, \tau}$ 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.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness