Underlying Set of Topological Space is Clopen

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Theorem

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

Then the underlying set $S$ of $T$ is both open and closed in $T$.


Proof

From the definition of topology, $S$ is open in $T$.


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

From Underlying Set of Topological Space is Closed $S$ is closed in $T$.

Hence the result.

$\blacksquare$


Sources