Connected components of a metric space are complete subspaces.

Question

If somene have just a hint for me i would be very grateful, because i don't know how to start. Let $A$ be a connected componente of $x \in M$,with $(M,d)$ metric space. Supose that $(x_n)_{n \in \mathbb{N} }$ is a cauchy sequence in $A$. I can't apply the definition of cauchy sequence, and can't use the fact that the connected component is a closed set.

Answer 1

Under the assumption that the whole space is complete, then it's only a matter of using these facts:

• every connected component is a closed subset;
• every closed subset of a complete metric space is also complete.

On the other hand, without assuming that the whole space is complete, then the statement is just plain false. Observe that if $M$ is connected, then its only connected component is $M$ itself, which may well not be complete.