Clopen Sets in Indiscrete Topology

Theorem

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

The only subsets of $S$ which are both closed and open in $T$ are $S$ and $\O$.

Proof

By definition of indiscrete topological space, the only open sets in $\struct {S, \tau}$ are $S$ and $\O$.

From Open and Closed Sets in Topological Space, both $S$ and $\O$ are both closed and open in $\struct {S, \tau}$ are $S$ and $\O$.

Hence the result.

$\blacksquare$

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets