If $\mathcal P$ is a topological property, then a space $X$ is called discretely $\mathcal P$ if the closure of every discrete subset of $X$ has $\mathcal P$. The property $\mathcal P$ is discretely reflexive in a class $\mathcal A$ if a space $X$ from $\mathcal A$ has $\mathcal P$ if and only if it is discretely $\mathcal P$. I proved in 1988 that compactness is discretely reflexive in the class of all spaces and it is still an open question whether the Lindel"of property is discretely reflexive. However, Arhangel’skii and Buzyakova proved in 1999 that the Lindel"of property is discretely reflexive in spaces of countable tightness. In this talk I will show that pseudocharacter is discretely reflexive in Lindel"of $\Sigma$-groups but countable tightness is not discretely reflexive in hereditarily Lindel"of spaces. Besides, I will present some results on discrete reflexivity of topological properties in spaces $C_p(X)$.