# Underlying Set of Topological Space is Clopen

Jump to navigation
Jump to search

## 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

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3.7$: Definitions: Examples $3.7.3 \ \text{(b)}$