Is there any relation between connectedness of metrizible spaces and their completeness

Question

I'm new to topology and apologize if my questions might be trivial.

I'm wondering about the causality relation between connected and complete metric spaces. I do wanna know whether the following statements are theorems: 1. If (X,T) is a connected metrizible topological space, then (X,d) (where d is the metric inducing T) is a complete metric space. 2. If (X,d) is a complete metric space, then the topological space (X,T) induced by (X,d) is connected.

Of course these two satements are the converse of each other.

Answer 1

The answer to both questions is negative. The space $\mathbb{R}\setminus(-1,1)$, with its usual metric, is complete, but disconnected. And $(-1,1)$, with its usual metric, is connected but not complete.