Topological Space is Connected iff any Proper Non-Empty Subset has Non-Empty Boundary
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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 $\partial S \ne \O$.
$\blacksquare$