Open and Closed Sets in Topological Space

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Theorem

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

Then $S$ and $\varnothing$ are both both open and closed in $T$.


Proof

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


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


From Empty Set is Element of Topology, $\varnothing$ is open in $T$.

From Empty Set is Closed in Topological Space, we have that $\varnothing$ is closed in $T$.

$\blacksquare$


Sources