Any two closed subset of metric space are connected iff they are disjoint

Any two closed subset of metric space are connected iff they are disjoint
 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.

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