If a topological space is separable are all discrete subspaces countable?

Question

Let $(X,\tau)$ be a topological space and $Y\subset X$ be such that $|Y|=|\mathbb{R}|$ and $(Y,\tau|_y)$ is a discrete space. Does it follow that $(X,\tau)$ is not separable?

What about the reciprocal, if $X$ is not separable can we always find a discrete uncountable subspace?

Answer 1

Let $\Psi$ be maximal family of almost disjoint subsets of $\mathbb{N}$ that has cardinality continuu. Consider the associated Mrówka space $X_\Psi = \mathbb{N} \cup \Psi$. Then $\Psi$ is a discrete subspace and $X_\Psi$ is separable because $\mathbb{N}$ is dense in it.