In 2008 Juhász and Szentmiklóssy established that for every compact space $X$ there exists a discrete $D\subset X\times X$ with $\left|D\right|=d\left(X\right)$. We generalize this result in two directions: the first one is to prove that the same holds for any Lindelöf $\Sigma$-space $X$ and hence $X^{\omega }$ is $d$-separable. We give an example of a countably compact space $X$ such that $X^{\omega }$ is not $d$-separable. On the other hand, we show that for any Lindelöf $p$-space $X$ there exists a discrete subset $D\subset X\times X$ such that $\Delta =\{(x,x):x\in X\}\subset \overline{D}$; in particular, the diagonal $\Delta$ is a retract of $\overline{D}$ and the projection...