Underlying Set of Topological Space is Clopen

Theorem

Let $T = \struct {S, \tau}$ 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 Underlying Set of Topological Space is Closed $S$ is closed in $T$.

Hence the result.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Examples $3.7.3 \ \text{(b)}$