Counterexample about Jones lemma with special weak condition.

Question

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 weaker 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 $2^{|D|} \le|S|$?

Answer 1

Yes, take, for example the Sorgenfrey plane $P$. A standard example of a non-normal space. It is separable and its anti-diagonal $\lbrace (x,-x):x\in\mathbb{R}\rbrace$ is closed and discrete, so Jones' Lemma is applicable in this case. Take any Hausdorff compactification of $P$; the result is a separable normal space and now the anti-diagonal is a relatively discrete subspace of the right cardinality.