I read a paper and met the concept Katetov extension. What is Katetov extension of the natural numbers? Reference on it are also welcome.
Katětov proved that any Hausdorff space $X$ can be densely embedded in an $H$-closed space; he did it by constructing a specific $H$-closed extension, $\kappa X$, of $X$. (A space is $H$-closed if it is closed in every Hausdorff space in which it is embedded.)
The first paper here, by Mukherjee, Sengupta, and Ghosh, treats some cardinal functions on $\kappa D$ for discrete spaces $D$.