Is the space $\mathbb N^ \mathbb N$ metrisable?

Question

Given the space $\mathbb N^ \mathbb N$ with the topology generated by basis sets of the form:

$$[V,n] = \{x \in \mathbb N^ \mathbb N ; V \text{ is an n prefix of x}\}$$

I can see that this space is separable.

My question is: is it metrisable? or even completely metrisable?

Thank you!