Equivalence of Definitions of Connected Topological Space/No Subsets with Empty Boundary implies No Clopen Sets

Theorem

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

Let $T$ be such that the only subsets of $S$ whose boundary is empty are $S$ and $\O$.

Then the only clopen sets of $T$ are $S$ and $\O$.

Proof

Let $H \subseteq S$ be a clopen set of $T$.

From Set is Clopen iff Boundary is Empty, $H$ has an empty boundary.

We have by hypothesis that $H = S$ or $H = \O$.

That is, the only clopen sets of $T$ are $S$ and $\O$.

$\blacksquare$