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I am not really sure if my ideas in this topic are correct. Can anyone help me?

  1. Finding the connected components of a metric space $X$.

Suppose there are two connected components $C_1, C_2$ of $X$. In this case, I have to prove that each component is connected, non-empty, open and closed in $X$, and that:

  • $C_1 \cap C_2 = \emptyset$
  • $C_1 \cup C_2 = X$

And in case there are more than 2, or infinitely many, what do I need to prove?

  1. Proving that a subset $S \subset X$ is a connected component.

As far as I know, there's no need now for a connected component to be an open set. Then, what do I need to prove exactly, appart from connectedness of $S$? I suppose that there's no other superset $T\supset S$ which is connected, but how?

  1. Not every connected component is a path-connected component, and not every path-connected component is a connected component.

For the second statement, are there any well-known examples?

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  • $\begingroup$ Just a comment: if you manage to prove that $C_1 \cap C_2 = \varnothing$ and $C_1 \cup C_2 = X$, then you do not need to show that $C_1$ and $C_2$ are open and closed. Open will imply closed, and vice versa, since you already know that they are complement in $X$. $\endgroup$
    – Didier
    Commented Jul 13 at 15:33
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    $\begingroup$ Any connected component is closed. If there are infinitely many components, it's not guaranteed that each of them is open. There are non-discrete spaces whose every connected component consists of a point ( like the Cantor set). $\endgroup$
    – orangeskid
    Commented Jul 13 at 17:38

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