Topological Space is Connected iff any Proper Non-Empty Subset has Non-Empty Boundary

Theorem

Let $\struct {X, \tau}$ be a topological space.

Then $\struct {X, \tau}$ is connected if and only if for each proper non-empty subset $S \subseteq X$, we have $\partial S \ne \O$.

Proof

From Connected iff no Proper Clopen Sets, we have that:

$\struct {X, \tau}$ is connected if and only if there exists no proper non-empty clopen set $S \subseteq X$.

From Set is Clopen iff Boundary is Empty, we have that:

$S \subseteq X$ is clopen if and only if $\partial S = \O$.

Hence we have:

$\struct {X, \tau}$ is connected if and only if there exists no proper non-empty set $S \subseteq X$ such that $\partial S = \O$.

That is:

$\struct {X, \tau}$ is connected if and only if for each proper non-empty set $S \subseteq X$, we have $\partial S \ne \O$.

$\blacksquare$