Sierpiński Space is T5

Theorem

Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.

Then $T$ is a $T_5$ space.


Proof

The only closed sets in $T$ are $\O$, $\set 1$ and $\set {0, 1}$.

So there are no two separated sets $A, B \subseteq \set {0, 1}$.

So $T$ is a $T_5$ space vacuously.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $11$. Sierpinski Space: $18$
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