In Willard's General Topology, (ed. Dover, p100), the lemma 15.2 (Jones' lemma) states that:
If $X$ contains a dense set $D$ and a closed, relatively discrete subspace $S$ with $|S| \ge 2^{|D|}$, then $X$ is not normal.
However, the proof leads momentarily to the condition $|P(S)|\gt|P(D)|$ that would forbid the normality, thus $|S|\gt|D|$, but this result is immediately inexplicably weakened into $|S|\ge2^{|D|}$.
But the previous form was both stronger and easier to manipulate, so why is it necessary to weaken it?
In general, $|S|>|D|$ does not imply $|P(S)|>|P(D)|$ (or at least ZFC cannot prove that it does). For instance, it is consistent with ZFC that $2^{\aleph_0}=2^{\aleph_1}=\aleph_2$. On the other hand, $|S|\geq 2^{|D|}$ definitely does imply $|P(S)|>|P(D)|$ since $$|P(S)|=2^{|S|}\geq 2^{2^{|D|}}>2^{|D|}=|P(D)|$$ with the strict inequality by Cantor's theorem.