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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,Â
​\( \bar{A}\cap B=\phi \)​
​\( \therefore A\cap B=\phi \)​ (​\( \because \)​ A is closed ​\( \implies A=\bar{A} \)​)Â
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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