Can there be a metric space where no contraction has a fixed point?

Question

We know that:

If $X$ is a metric space, then every contraction has at most one fixed point.

(Note: if metric space is complete, then we have existence and uniqueness)

I wonder if there can be a metric space for which no contraction has a fixed point. Thanks.

Answer 1

If $X$ is non-empty then we can choose some $x_0\in X$ and define $f(x)=x_0$ for all $x\in X$. This is a contraction with a fixed point.