Empty Set is Closed in Topological Space

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Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Then $\O$ is closed in $T$.


Proof

From the definition of closed set, $U$ is open in $T = \struct {S, \tau}$ if and only if $S \setminus U$ is closed in $T$.

From Underlying Set of Topological Space is Clopen, $S$ is open in $T$.

From Set Difference with Self is Empty Set, we have $S \setminus S = \O$, so $\O$ is closed in $T$.

$\blacksquare$


Sources