Is a complete metric space a continuum?

Question

By definition, a continuum is a compact connected metric space. Is it correct to say that a complete metric space is always compact connected? In other words, can I say that a complete metric space is a continuum? Thanks in advance

Answer 1

As Ittay Weiss pointed out, a complete metric space need not be compact: $\mathbb R$ is a counterexample. In fact, the implication works the other way round: all compact metric spaces are complete! (The standard theorem I'm quoting says that a metric space is compact iff it is totally bounded.)

A complete metric space need not be connected either. For example, $[0,1] \cup [2,3]$ is complete (since it is a closed subspace of the complete space $\mathbb R$), but it is not connected.