Cofinality of a cardinal

Question

What can be the cofinality of this cardinal: $$\lambda^{\aleph_0}$$

where $\lambda$ is an infinite cardinal ? Can the cofinality be countable ?

Answer 1

It cannot be countable by the following theorem about cofinality:

For any infinite cardinal $\kappa$ we have $\kappa<\kappa^{\mathrm{cf}(\kappa)}$.

If $\kappa=\lambda^{\aleph_0}$ and $\mathrm{cf}(\kappa)=\aleph_0$, then we get a contradiction

$$\kappa<\kappa^{\mathrm{cf}(\kappa)}=(\lambda^{\aleph_0})^{\aleph_0}=\lambda^{\aleph_0\cdot\aleph_0}=\lambda^{\aleph_0}=\kappa$$

Other than that, there's not much to say without knowing more about $\lambda$.