Jones Lemma is One scale about recognizing that a topological space is not normal.
This lemma tells us, if The topological space $X$ has a dense subset $D$ and a closed discrete subset $S$, with the property that $2^{|D|} \le|S|$, it couldn't be a normal space. But I think there is no apparent counterexample about the weeker condition of this lemma as follows.

Q. Is there an example of normal space $X$ which has a dense subset $D$ and a discrete subset $S$ with the property that $2^{|D|} \le|S|$?