Theorem:
Any two closed subset of metric space are connected iff they are disjoint.
Proof:
Let (X, d) be a metric space and A & B are any two closed subsets of X.
Let A & B are separated sets.
Claim: A & B are disjoint.
As A & B are separated,
\( \bar{A}\cap B=\phi \)
\( \therefore A\cap B=\phi \) (\( \because \) A is closed \( \implies A=\bar{A} \))
Conversely,
Suppose that A & B are disjoint.
Claim: A & B are separated.
By hypothesis,
\( A\cap B=\phi \)
Since, A is closed, \( A=\bar{A} \)
\( \implies \bar{A}\cap B=\phi \) ———–(1)
Similarly, B is closed, \( B=\bar{B} \)
\( \implies A\cap \bar{B}=\phi \) ———–(2)
From (1) & (2),
A and B are separated sets.
