Show that the components of a metric space is either identical or pairwise disjoint.
Let's prove the statement:
Suppose we have a metric space \( (X, d) \), where \( X \) is the set and \( d \) is the metric. Let \( A \) and \( B \) be two distinct components of \( X \). We want to show that \( A \cap B = \emptyset \).
By definition, a component of a metric space is a maximal connected subset. Therefore, \( A \) and \( B \) are both connected subsets of \( X \).
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