Arens-Fort Space is Completely Normal
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Theorem
Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a completely normal space.
Consequently, $T$ satisfies all weaker separation axioms.
Proof
We have:
and so by definition $T$ is completely normal.
$\blacksquare$
See Sequence of Implications of Separation Axioms for confirmation of the statement about weaker separation axioms.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $26$. Arens-Fort Space: $2$