Characterizations of paracompactness and Lindelöfness by the separation property

The separation property in our title is that, for two spaces $X$ and $Y$, any two disjoint closed copies of $X$ in $Y$ are separated by open sets in $Y$. It is proved that a Tychonoff space $X$ is paracompact if and only if this separation property holds for the space $X$ and every Tychonoff space $Y$ which is a perfect image of $X\times \beta X$ (where $\beta X$ denotes the Stone-Cech compactification of $X$). Moreover, we give a characterization of Lindelöfness in a similar way under the assumption of paracompactness.