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Q: What is the definition of a component of a metric space?

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Equivalent definitions:

1. A connected subspace of a topological space T is said to be a component of T if it is not properly contained in any connected subspace of T.
2. A maximal connected set of T that contains a given point of T.
3. Let \~ be the equivalence relation given by considering two points of T equivalent if there exists a connected subset of T containing both points. Then the elements of T/\~ are the components of T.

And yes, *component* and *connected component* are synonyms in this context.
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I am not very proficient in topology so I might be wrong, but what you may be asking for are "maximal connected subsets", that is, connected subsets whose every proper superset is not connected.

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