Discrete subspace of $\mathbb{N}^\mathbb{N}$ Endow $\mathbb{N}$ with the discrete topology. Does $\mathbb{N}^\mathbb{N}$ contain a discrete subspace of size $2^{\aleph_0}$?

Since $\mathbb{N}$ is separable (with the discrete topology), we have that $\mathbb{N}^{{\mathbb{N}}}$ is separable with the product topology. Therefore, any subspace is separable, so there is no discrete subspace of cardinal $2^{\aleph_{0}}$