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

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

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

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

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

Then by definition $S \setminus H$ is also clopen.

Hence, both $H$ and $S \setminus H$ are their closures, and:


 * $\partial H = H^- \cap \left({S \setminus H}\right)^- = H \cap \left({S \setminus H}\right) = \varnothing$

Then $H$ has an empty boundary.

By hypothesis, $H = S$ or $H = \varnothing$.

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